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Values Of Trigonometric Functions At Sum Of Difference Of Angles

Class 11th Mathematics RD Sharma Solution
Exercise 7.1
  1. If sinA = 4/5 And cosB = 5/13, where 0 A, B π/2, find the values of the following: (i)…
  2. If SinA = 12/13 And sinB = 4/5, where π/2A π And 0 B π/2, find the following: (i) sin(A…
  3. If sinA = 3/5, cosB = -12/13, where A And B Both lie in second quadrant, find the value of…
  4. If cosA = - 24/25 And cosB = 3/5, where π A 3π/2 And 3π/2 B 2π, find the following: (i)…
  5. If tanA = 3/4, cosB = 9/41, where πA 3π/2 And 0 B π/2, find tan(A +B).…
  6. If sinA = 1/2, cosB = 12/13, where π/2A π And 3π/2 B 2π, find tan(A -B).…
  7. If SinA = 1/2, cosB = , where π/2A π And 0 B π/2, find the following: (i) tan(A +B)(ii)…
  8. Evaluate the following: (i) sin 78^0 cos 18^0 - cos 78^0 sin 18^0 (ii) cos 47^0 cos 13^0 -…
  9. If cosA = -12/13 and cotB = 24/7, where A lies in the second quadrant and B in the third…
  10. Prove that: cos 7π/12 + cos π/12 = sin 5π/12 - sin π/12
  11. Prove that:
  12. (i) (ii) (ii) Prove that:
  13. Prove that:
  14. Prove that:
  15. Prove that:
  16. Prove that:
  17. If tanA = 5/6 And tanB = 1/11, prove thatA +B = π/4.
  18. If tanA = m/m-1 And tanB = 1/2m - 1, then prove that A -B = π/4.
  19. cos^2 π/4 - sin^2 prove that:
  20. sin^2 (n + 1)A - sin^2 nA = sin(2n + 1)A sinA prove that:
  21. Prove that:
  22. Prove that:
  23. Prove that:
  24. sin^2 B = sin^2 A + sin^2 (A-B) - 2sinA cosB sin(A-B) Prove that:…
  25. cos^2 A + cos^2 B - 2 cosA cosB cos(A +B) = sin^2 (A +B) Prove that:…
  26. Prove that:
  27. tan 8x - tan6x - tan 2x = tan 8x tan 6x tan 2x Prove that:
  28. Prove that:
  29. tan 36^0 + tan 9^0 + tan 36^0 tan 90 = 1 Prove that:
  30. tan13x - tan 9x - tan 4x = tan 13 x Tan 9x tan 4x Prove that:
  31. Prove that:
  32. If , show that .
  33. If tanA = tanB, prove that .
  34. If tan(A +B) = And tan(A -B) = y, find the values of tan 2A And tan 2B.…
  35. If CosA + SinB = m And SinA + CosB = n, prove that 2 Sin(A +B) = m^2 + n^2 - 2.…
  36. If tanA + tanB =A And CotA + CotB =B, prove that: cot(A +B) = 1/a - 1/b.…
  37. If lies in the first quadrant And cos x = 8/17, then prove that
  38. If then prove that .
  39. If sin(α + β) = 1 And sin(α - β) = 1/2, where , then find the values of tan(α + 2β) And…
  40. If α,β are two different values of x lying between 0 And 2π which satisfy the equation 6…
  41. If sin α + sin β =A And cos α + cos β =B, show that (i) (ii)
  42. Prove that:
  43. Prove that:
  44. Prove that:
  45. If sin α sin β - cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.…
  46. If tan α = x + 1, tanβ = x - 1, show that 2 cot(α - β) = x^2 .
  47. IfAngle θ is divided into two parts such that the tangents of the one part is λ times the…
  48. If , then show that sin α + cos α = cos x.
  49. If α And β are two solutions of the equation Atanx + Bsecx = c, then find the values of…
Exercise 7.2
  1. Find the maximum and minimum values of each of the following trigonometrical…
  2. Reduce each of the following expressions to the Sine And Cosine of A single…
  3. cos x - sin x Reduce each of the following expressions to the Sine And Cosine…
  4. 24 cos x + 7 sin x Reduce each of the following expressions to the Sine And…
  5. Show that Sin 100^0 - Sin 10^0 is positive.
  6. Prove that (2 root 3+3) sinx+2 root 3 cosx lies between - (2 root 3 + root 15)…
Very Short Answer
  1. If α + β - γ = π, and sin2 α + sin2 β – sin2 γ = λ sin α sin β cos γ, then write the value…
  2. If xcostheta = ycos ( theta + { 2 pi }/{3} ) = zcos ( theta + frac { 4 pi }/{3} )…
  3. Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
  4. Write the maximum values of 12 sin x – 9 sin2 x.
  5. If 12 sin x – 9 sin2 x attains its maximum value at x = α, then write the value of sin…
  6. Write the interval in which the values of 5cosx+3cos ( x + { pi }/{3} ) + 3 lie.…
  7. If tan (A + B) = p and tan (A – B) = q, then write the value of tan 2B.…
  8. If { cos (x-y) }/{ cos (x+y) } = frac {m}/{n} then write the value of tan x tan y.…
  9. If a = bcos { 2 pi }/{3} = cos frac { 4 pi }/{3} then write the value of ab + bc +…
  10. If A + B = C, then write the value of tan A tan B tan C.
  11. If sin α – sin β = a and cos α + cos β = b, then write the value of cos (α + β).…
  12. If tanalpha = {1}/{ 1+2^{-x} } and tanbeta = {1}/{ 1+2^{x+1} } then write…
Mcq
  1. The value of i sin^{2} { 5 pi }/{12} - sin^{2} frac { pi }/{12} s Mark the correct…
  2. If A + B + C = π, then sec A (cos B cos C – sin B sin C) is equal to Mark the correct…
  3. tan 20o + tan 40o + √3 tan 20o tan 40o is equal to Mark the correct alternative in the…
  4. If tana = {a}/{a+1} and tanb = {1}/{2a+1} then the value of A + B is Mark the…
  5. If 3 sin x + 4 cos x = 5, then 4 sin x – 3 cos x = Mark the correct alternative in the…
  6. If in a ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C = Mark the correct…
  7. tan 3A – tan 2A – tan A is equal to Mark the correct alternative in the following:…
  8. If a+b+c = pi then {tana+tanb+tanc}/{tanatanbtanc} is equal to Mark the correct…
  9. If cosp = {1}/{7} and cosq = {13}/{14} where P and Q both are acute angles.…
  10. If cot (α + β) = 0, then sin (α + 2 β) is equal to Mark the correct alternative in the…
  11. { cos10^{degree } + sin10^{circ} }/{ cos10^{circ} - sin10^{circ} } is equal to Mark…
  12. The value of cos^{2} ( { pi }/{6} + x ) - sin^{2} ( frac { pi }/{6} - x ) is Mark…
  13. If tantheta _{1} tantheta_{2} = k then { cos ( theta _{1} - theta_{2} ) }/{…
  14. If sin (π cos x) = cos (π sin x), then sin 2x = Mark the correct alternative in the…
  15. If tantheta = {1}/{2} and tanphi = {1}/{3} then the value of θ + ϕ is Mark…
  16. The value of cos (36o – A) cos (36o + A) + cos (54o + A) cos (54o – A) is Mark the…
  17. If tan (π/4 + x) + tan (π/4 – x) = a, then tan2 (π /4 + x) + tan2 (π /4 – x) = Mark…
  18. If tan (a-b) = 1 , sec (a+b) = {2}/{ root {3} } then the smallest positive value…
  19. If A – B = π /4, then (1 + tan A) (1 – tanB) is equal to Mark the correct alternative…
  20. The maximum value of sin^{2} ( { 2 pi }/{3} + x ) + sin^{2} ( frac { 2 pi }/{3} -…
  21. If cos (a-b) = {3}/{5} and tan A tan B = 2, then Mark the correct alternative in…
  22. If tan 69o + tan 66o – tan 69o tan 66o = 2k, then k = Mark the correct alternative in…
  23. If tanalpha = {x}/{x+1} and tanbeta = {1}/{2x+1} then α + β is equal to…
Exercise 7.1
Question 1.

If sinA = 4/5 And cosB = 5/13, where 0 <A, B < π/2, find the values of the following:

(i) sin(A +B)

(ii) cos(A +B)

(iii) sin(A –B)

(iv) cos(A -B)
Answer:

Given sinA = 4/5 And cosB = 5/13


We know that where 0 <A,B < π/2






Then,


(i) Sin(A +B)


We know that sin(A +B) = sinA cosB + cosA sinB





(ii) Cos(A +B)


We know that cos(A +B) = cosA cosB-sinA sinB





(iii) Sin(A –B)


We know that sin(A -B) = sinA cosB - cosA sinB





(iv) Cos(A –B)


We know that cos(A -B) = cosA cosB+sinA sinB
Question 2.

If SinA = 12/13 And sinB = 4/5, where π/2<A < π And 0 <B < π/2, find the following:

(i) sin(A +B) (ii) cos(A +B)
Answer:

Given sinA = 12/13 And sinB = 4/5 where π/2 <A < π And 0 <B < π/2


We know that






(i) Sin(A +B)


We know that sin(A +B) = sinA cosB + cosA sinB





(ii) Cos(A +B)


We know that cos(A +B) = cosA cosB-sinA sinB
Question 3.

If sinA = 3/5, cosB = –12/13, where A And B Both lie in second quadrant, find the value of sin(A +B).
Answer:

Given sinA = 3/5 And cosB = -12/13


A And B lie in the second quadrant.


So sine function is positive And cosine function is negative.


We know that






Now consider sin(A +B),


⇒ sin(A +B)





Question 4.

If cosA = – 24/25 And cosB = 3/5, where π <A < 3π/2 And 3π/2 <B < 2π, find the following:

(i) sin(A +B) (ii) cos(A +B)
Answer:

Given cosA = -24/25 And cosB = 3/5 where π <A < 3π/2 And 3π/2 <B < 2π


A is in third quadrant And B is in fourth quadrant.


Here, sine function is negative.


We know that






Then,


(i) Sin(A +B)


We know that sin(A +B) = sinA cosB + cosA sinB






(ii) Cos(A +B)


We know that cos(A +B) = cosA cosB-sinA sinB
Question 5.

If tanA = 3/4, cosB = 9/41, where π<A < 3π/2 And 0 <B < π/2, find tan(A +B).
Answer:

Given tanA = 3/4 And cosB = 9/41 where π <A < 3π/2 And 0 <B < π/2


A is in third quadrant And B is in first quadrant.


Tan function And sine function are positive.


We know that






We know that




We know that






Question 6.

If sinA = 1/2, cosB = 12/13, where π/2<A < π And 3π/2 <B < 2π, find tan(A -B).
Answer:

Given sinA = 1/2 And cosB = 12/13 where π/2 <A < π And 3π/2 <B < 2π


A is in second quadrant And B is in fourth quadrant.


In the second quadrant, the sine function is positive And cosine And tan functions negative.


In the fourth quadrant, sine And tan functions are negative, And cosine function are positive.


We know that







We know that
Question 7.

If SinA = 1/2, cosB = , where π/2<A < π And 0 <B < π/2, find the following:

(i) tan(A +B)(ii) tan(A -B)
Answer:

Given sinA = 1/2 And cosB = √3/2 where π/2 <A < π And 0 <B < π/2,


A is in second quadrant And B is in first quadrant.


In the second quadrant, the sine function is positive And cosine And tan functions are negative.


In first quadrant, All functions are positive.


We know that







Then,


(i) tan(A +B)


We know that




= 0


(ii) tan(A –B)


We know that






Question 8.

Evaluate the following:

(i) sin 780 cos 180 – cos 780 sin 180 (ii) cos 470 cos 130 - sin 470 sin 130

(iii) sin 360 cos 90 + cos 360 sin 90 (iv) cos 800 cos 200 + sin 800 sin 200
Answer:

(i) Given sin 780 cos 180 – cos 780 sin 180


We know that sin(A -B) = sinA cosB - cosA sinB


⇒ sin 780 cos 180 – cos 780 sin 180 = sin(78 – 18) °


= sin 60°




(ii) Given cos 470 cos 130 - sin 470 sin 130


We know that cosA cosB – sinA sinB = cos(A +B)


⇒ cos 470 cos 130 - sin 470 sin 130 = cos(47 + 13) °


= cos 60°


= 1/2


∴ cos 470 cos 130 - sin 470 sin 130 = 1/2


(iii) Given sin 360 cos 90 + cos 360 sin 90


We know that sin(A +B) = sinA cosB + cosA sinB


⇒ sin 360 cos 90 + cos 360 sin 90 = sin(36 + 9) °


= sin 45°




(iv) Given cos 800 cos 200 + sin 800 sin 200


We know that cosA cosB + sinA sinB = cos(A -B)


⇒ cos 800 cos 200 + sin 800 sin 200 = cos(80 - 20) °


= cos 60°


= 1/2


∴ cos 800 cos 200 + sin 800 sin 200 = 1/2
Question 9.

If cosA = –12/13 and cotB = 24/7, where A lies in the second quadrant and B in the third quadrant, find the values of the following:

(i) sin(A +B) (ii) cos(A +B) (iii) tan(A +B)
Answer:

Given cosA = -12/13 And cotB = 24/7


A lies in second quadrant And B in the third quadrant.


The sine function is positive in the second quadrant and in the third quadrant,Both sine And cosine functions are negative.


We know that







Now,


(i) Sin(A +B)


We know that sin(A +B) = sinA cosB + cosA sinB





(ii) Cos(A +B)


We know that cos(A +B) = cosA cosB-sinA sinB





(iii) Tan(A +B)


We know that





Question 10.

Prove that: cos 7π/12 + cos π/12 = sin 5π/12 – sin π/12
Answer:

⇒ 7π/12 = 105°, π/12 = 15°; 5π/12 = 75°


LHS = cos 105° + cos 15°


= cos(90° + 15°) + sin(90° - 75°)


= -sin 15° + sin 75°


= sin 75° - sin 15° = RHS


Hence proved.
Question 11.

Prove that:


Answer:

LHS




We know that sin(A ±B) = sinA cosB ± cosA sinB


= RHS


Hence, proved.
Question 12.

Prove that:

(i) (ii)

(ii)


Answer:

(i) LHS


Dividing numerator And denominator by cos 11°,





We know that



= tan 56° = RHS


Hence proved.


(ii) LHS


Dividing numerator And denominator by cos 9°,





We know that



= tan 54° = RHS


Hence proved.


(iii) LHS


Dividing numerator And denominator by cos 8°,





We know that



= tan 37° = RHS


Hence proved.
Question 13.

Prove that:
Answer:

We know that sin(A +B) = sinA cosB + cosA sinB





= sin 90°


= 1 = RHS


Hence proved.
Question 14.

Prove that:
Answer:

We know that sin(A -B) = sinA cosB - cosA sinB





= sin 60°


= RHS


Hence, proved.
Question 15.

Prove that:
Answer:

We know that sin(A +B) = sinA cosB + cosA sinB





= sin 90°


= 1 = RHS


Hence proved.
Question 16.

Prove that:


Answer:

We know that


HereA = 69° And B = 66°



= tan 135°


= - tan 45°


= -1 = RHS


Hence proved.
Question 17.

If tanA = 5/6 And tanB = 1/11, prove thatA +B = π/4.
Answer:

Given


We know that





= 1


⇒ tan(A +B) = tan π/4


∴A + B = π/4


Hence proved.
Question 18.

If tanA = m/m–1 And tanB = 1/2m – 1, then prove that A –B = π/4.
Answer:

Given


We know that





= 1


⇒ tan(A –B) = tan π/4


∴A –B = π/4


Hence proved.
Question 19.

prove that:

cos2 π/4 - sin2
Answer:

LHS


We know that cos2A – sin2B = cos(A +B) cos(A –B)






= RHS


Hence, proved.
Question 20.

prove that:

sin2(n + 1)A – sin2nA = sin(2n + 1)A sinA
Answer:

We know that sin2A – sin2B = sin(A +B) sin(A –B)


HereA =(n + 1)A And B = nA


⇒ LHS: sin2(n + 1)A – sin2nA = sin((n + 1)A + nA) sin((n + 1)A – nA)


= sin(nA +A + nA) sin(nA +A – nA)


= sin(2nA +A) sin(A)


= sin(2n + 1)A sinA = RHS


Hence proved.
Question 21.

Prove that:
Answer:

LHS


We know that sin(A ±B) = sinA cosB ± cosA sinB And cos(A ±B) = cosA cosB∓sinA sinB




= tanA = RHS


Hence proved.
Question 22.

Prove that:
Answer:

LHS


We know that sin(A –B) = sinA cosB – cosA sinB




= tanA – tanB + tanB – tan C + tan C – tanA


= 0 = RHS


Hence proved.
Question 23.

Prove that:
Answer:

LHS


We know that sin(A –B) = sinA cosB – cosA sinB




= cotB – cotA + cot C – cotB + cotA – cot C


= 0 = RHS


Hence proved.
Question 24.

Prove that:

sin2B = sin2A + sin2(A-B) – 2sinA cosB sin(A-B)
Answer:

RHS = sin2A + sin2(A -B) – 2 sinA cosB sin(A -B)


= sin2A + sin(A -B) [sin(A –B) – 2 sinA cosB]


We know that sin(A –B) = sinA cosB – cosA sinB


= sin2A + sin(A -B) [sinA cosB – cosA sinB – 2 sinA cosB]


= sin2A + sin(A -B) [-sinA cosB – cosA sinB]


= sin2A - sin(A -B) [sinA cosB + cosA sinB]


We know that sin(A +B) = sinA cosB + cosA sinB


= sin2A – sin(A –B) sin(A +B)


= sin2A – sin2A + sin2B


= sin2B = LHS


Hence proved.
Question 25.

Prove that:

cos2A + cos2B – 2 cosA cosB cos(A +B) = sin2(A +B)
Answer:

LHS = cos2A + cos2B – 2 cosA cosB cos(A +B)


= cos2A + 1 – sin2B - 2 cosA cosB cos(A +B)


= 1 + cos2A – sin2B - 2 cosA cosB cos(A +B)


We know that cos2A – sin2B = cos(A +B) cos(A –B)


= 1 + cos(A +B) cos(A –B) - 2 cosA cosB cos(A +B)


= 1 + cos(A +B) [cos(A –B) – 2 cosA cosB]


We know that cos(A -B) = cosA cosB + sinA sinB.


= 1 + cos(A +B) [cosA cosB + sinA sinB – 2 cosA cosB]


= 1 + cos(A +B) [-cosA cosB + sinA sinB]


= 1 - cos(A +B) [cosA cosB - sinA sinB]


We know that cos(A +B) = cosA cosB - sinA sinB.


= 1 – cos2(A +B)


= sin2(A +B) = RHS


Hence proved.
Question 26.

Prove that:
Answer:

LHS


We know that




We know that (x + y)(x – y) = x2 – y2



Hence, proved.
Question 27.

Prove that:

tan 8x – tan6x – tan 2x = tan 8x

tan 6x tan 2x
Answer:

We have 8x = 6x + 2x


⇒ tan 8x = tan(6x + 2x)


We know that



⇒ tan8x (1 – tan6x tan2x) = tan6x + tan2x


⇒ tan8x – tan8x tan 6x tan2x = tan6x + tan2x


∴ tan8x – tan6x – tan2x = tan8x tan6x tan2x


Hence, proved.
Question 28.

Prove that:
Answer:

⇒ π/12 = 15° And π/6 = 30°


We have 15° + 30° = 45°


⇒ tan(15° + 30°) = tan 45°


We know that



⇒ tan15° + tan30° = 1 – tan15° tan30°


∴ tan15° + tan30° + tan15° tan30° = 1


Hence, proved.
Question 29.

Prove that:

tan 360 + tan 90 + tan 360 tan 90 = 1
Answer:

We have 36° + 9° = 45°


⇒ tan(36° + 9°) = tan 45°


We know that



⇒ tan 36° + tan 9° = 1 – tan 36° tan 9°


∴ tan 36° + tan 9° + tan 36° tan 9° = 1


Hence proved.
Question 30.

Prove that:

tan13x – tan 9x – tan 4x = tan 13 x

Tan 9x tan 4x
Answer:

We have 13x = 9x + 4x


⇒ tan 13x = tan(9x + 4x)


We know that



⇒ tan 13x(1 – tan 9x tan 4x) = tan 9x + tan 4x


⇒ tan 13x – tan 13x tan 9x tan 4x = tan 9x + tan 4x


∴ tan 13x – tan 9x – tan 4x = tan 13x tan 9x tan 4x


Hence proved.
Question 31.

Prove that:


Answer:

LHS



⇒ tan3x = tan(2x + x) And tan x = tan(2x – x)




= tan3x tanx = RHS


Hence, proved.
Question 32.

If , show that .
Answer:

Given


LHS


We know that sin(A ±B) = sinA cosB ± cosA sinB And cos(A ±B) = cosA cosB∓sinA sinB




= tanA = RHS
Question 33.

If tanA = tanB, prove that .
Answer:

Given tanA = x tanB


LHS


We know that sin(A ±B) = sinA cosB ± cosA sinB



Dividing numerator And denominator by cosA cosB,






= RHS


Hence, proved.
Question 34.

If tan(A +B) = And tan(A -B) = y, find the values of tan 2A And tan 2B.
Answer:

Given tan(A +B) = x And tan(A –B) = y


Consider tan 2A = tan(A +A)


= tan(A +B +A –B)


We know that




Consider tan 2B = tan(B +B)


= tan(B +A +B –A)



We know that tan(-θ) = - tan θ
Question 35.

If CosA + SinB = m And SinA + CosB = n, prove that 2 Sin(A +B) = m2 + n2 – 2.
Answer:

Given cosA + sinB = m And sinA + cosB = n


RHS = m2 + n2 – 2


=(cosA + sinB)2 +(sinA + cosB)2 – 2


= cos2A + sin2B + 2 cosA sinB + sin2A + cos2B + 2 sinA cosB – 2


= 1 + 1 + 2(cosA sinB + sinA cosB) - 2


We know that sin(A -B) = sinA cosB - cosA sinB


= 2 sin(A +B)


= LHS


Hence, proved.
Question 36.

If tanA + tanB =A And CotA + CotB =B, prove that: cot(A +B) = 1/a – 1/b.
Answer:

Given tanA + tanB =A And cotA + cotB =B


Consider cotA + cotB =B




Then,


RHS





We know that


= cot(A +B) = LHS


Hence, proved.
Question 37.

If lies in the first quadrant And cos x = 8/17, then prove that
Answer:

Given x lies in the first quadrant i.e. 0 < x < π/2 And cos x = 8/17


We know that



LHS


= cos(30 + x) + cos(45 – x) + cos(120 – x)


We know that cos(A ±B) = cosA cosB∓sinA sinB


= cos 30° cos x – sin 30° sin x + cos 45° cos x + sin 45° sin x + cos 120° cos x + sin 120° sin x


= cos x(cos 30° + cos 45° + cos 120°) + sin x(-sin 30° + sin 45° + sin 120°)





= RHS


Hence, proved.
Question 38.

If then prove that .
Answer:

Given


We know that







Hence, proved.
Question 39.

If sin(α + β) = 1 And sin(α – β) = 1/2, where , then find the values of tan(α + 2β) And tan(2α + β).
Answer:

Given sin(α + β) = 1 And sin(α – β) = 1/2


⇒ α + β = 90° …(1) And α - β = 30° …(2)


Adding(1) And(2),


⇒ 2α = 120°


∴ α = 60°


Subtracting(2) from(1),


⇒ 2β = 60°


∴ β = 30°


Then,


∴ tan(α + 2β) = tan(60° + 2 × 30°) = tan 120° = -√3


And tan(2α + β) = tan(2 × 60° + 30°) = tan 150° = -(1/√3)
Question 40.

If α,β are two different values of x lying between 0 And 2π which satisfy the equation 6 cos x + 8 sin x = 9, find the value of Sin(α+β).
Answer:

Given 6 cos x + 8 sin x = 9


Case 1:


⇒ 6 cos x = 9 – 8 sin x


Squaring on both sides,


⇒ 36 cos2 x =(9 – 8 sin x)2


We know that cos2 x = 1 – sin2 x.


⇒ 36(1 – sin2 x) = 81 + 64 sin2 x – 144 sin x


⇒ 100 sin2 x – 144 sin x + 45 = 0


∴ cos α And cos β are the roots of the a bove equation


⇒ sin α sin β = 45/100


Case 2:


⇒ 8 sin x = 9 – 6 cos x


Squaring on both sides,


⇒ 64 sin2 x =(9 – 6 cos x)2


We know that sin2 x = 1 – cos2 x


⇒ 64(1 – cos2 x) = 81 + 36 cos2 x – 108 cos x


⇒ 100 cos2 x – 108 cos x + 17 = 0


∴ sin α And sin β are the roots of theAbove equation


⇒ cos α cos β = 17/100


Consider cos(α + β),


We know that cos(A +B) = cosA cosB-sinA sinB



We know that







Question 41.

If sin α + sin β =A And cos α + cos β =B, show that

(i)

(ii)


Answer:

Given sin α + sin β =A And cos α + cos β =B.


⇒A2 +B2 =(sin α + sin β)2 +(cos α + cos β)2


= sin2 α + sin2 β + 2 sin α sin β + cos2 α + cos2 β + 2 cos α cos β


= sin2 α + cos2 α + sin2 β + cos2 β + 2(sin α sin β + cos α cos β)


We know that cos(A -B) = cosA cosB+sinA sinB


∴A2 +B2 = 2 + 2 cos(α – β) …(1)


Then,


⇒B2 –A2 =(cos α + cos β)2 –(sin α + sin β)2


= cos2 α + cos2 β + 2 cos α cos β –(sin2 α + sin2 β + 2 sin α sin β)


=(cos2 α – sin2 β) +(cos2 β – sin2 α) – 2cos(α + β)


= 2 cos(α + β) cos(α – β) + 2 cos(α + β)


= cos(α + β)(2 + 2 cos(α – β)) …(2)


From(1) And(2),


⇒B2 –A2 = cos(α + β)(A2 +B2)


…(ii)


And




…(i)
Question 42.

Prove that:
Answer:

RHS




We know that sin(A -B) = sinA cosB - cosA sinB





= LHS


Hence, proved.
Question 43.

Prove that:
Answer:

RHS




We know that cos(A +B) = cosA cosB-sinA sinB





= LHS


Hence, proved.
Question 44.

Prove that:
Answer:

RHS




We know that sin(A -B) = sinA cosB - cosA sinB





= LHS


Hence, proved.
Question 45.

If sin α sin β - cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
Answer:

Given sin α sin β – cos α cos β + 1 = 0


⇒ -(cos α cos β – sin α sin β) + 1 = 0


We know that cos(A +B) = cosA cosB-sinA sinB


⇒ -cos(α + β) + 1 = 0


⇒ cos(α + β) = 1


We know that sin θ = √(1 – cos2 θ)


∴ sin(α + β) = 0 …(1)


Consider 1 + cot α tan β,




We know that sin(A ±B) = sinA cosB ± cosA sinB



= 0 = RHS


Hence, proved.
Question 46.

If tan α = x + 1, tanβ = x - 1, show that 2 cot(α - β) = x2.
Answer:

Given tan α = x + 1 And tan β = x – 1


LHS = 2 cot(α – β)


We know that





= x2 = RHS


Hence, proved.
Question 47.

IfAngle θ is divided into two parts such that the tangents of the one part is λ times the tangent of other, And ϕ is their difference, then show that .
Answer:

Let α And β be the two parts of angle θ.


Then, given θ = α + β And ϕ = α - β


Consider tan α = λ tan β



Applying componendo And dividendo,





We know that sin(A ±B) = sinA cosB ± cosA sinB





Hence, proved.
Question 48.

If , then show that sin α + cos α = cos x.
Answer:

Given


Dividing numerator And denominator on RHSBy cos α,




We know that





Consider sin α + cos α,



We know that sin(A +B) = sinA cosB + cosA sinB And cos(A +B) = cosA cosB-sinA sinB





= √2 cos x


∴ sin α + cos α = √2 cos x


Hence proved.
Question 49.

If α And β are two solutions of the equation Atanx + Bsecx = c, then find the values of sin(α + β).
Answer:

Given equation Atanx + Bsecx = c


⇒ c – Atanx = Bsecx


Squaring onBoth sides,


⇒(c – Atanx)2 = (Bsecx)2


⇒ c2 + A2tan2 x – 2actan x =B2sec2 x


⇒ c2 +A2 tan2 x – 2ac tan x = B2(1 + tan2 x)


⇒(a2 –B2) tan2 x – 2ac tan x +(c2 –B2) = 0


There are two solutions tan α And tan β in this quadratic.



We know that





Exercise 7.2
Question 1.

Find the maximum and minimum values of each of the following trigonometrical expressions:

(i) 12 sin x- 5 cos x

(ii) 12 cos x + 5 sin x+ 4

(iii)

(iv) sin x – cos x + 1
Answer:

We know that the maximum value of Acosα + Bsinα + c is


c + √(A2 +B2)


And the minimum value is c - √(a2 +B2).


(i) Given f(x) = 12 sin x – 5 cos x


Here A = -5,B = 12 and c = 0





⇒ -13 ≤ 12 sin x - 5 cos x ≤ 13


Hence, the maximum and minimum values of f(x) are 13 and -13 respectively.


(ii) Given f(x) = 12 cos x + 5 sin x + 4


Here A = 12,B = 5 and c = 4





⇒ -9 ≤ 12 cos x + 5 sin x + 4 ≤ 17


Hence, the maximum And minimum values of f(x) are 17 And -9 respectively.


(iii) Given


We know that sin(A -B) = sinA cosB - cosA sinB





Here





⇒ -3 ≤ ≤ 11


Hence, the maximum And minimum values of f(x) are 11 And -3 respectively.


(iv) Given f(x) = sin x – cos x + 1


Here A = -1,B = 1 And c = 1





Hence, the maximum And minimum values of f(x) are And respectively.
Question 2.

Reduce each of the following expressions to the Sine And Cosine of A single expression:
Answer:

Let f(x) = √3 sin x – cos x


Dividing and multiplying by √(3 + 1) = 2,



Sine of expression:



We know that sinA cosB – cosA sinB = sin(A –B)



Cosine of the expression:



We know that cosA cosB – sinA sinB = cos(A +B)
Question 3.

Reduce each of the following expressions to the Sine And Cosine ofA single expression:

cos x – sin x
Answer:

Let f(x) = cos x – sin x


Dividing and multiplying by √(1 + 1) = √2,



Sine of expression:



We know that sinA cosB – cosA sinB = sin(A –B)



Cosine of the expression:



We know that cosA cosB – sinA sinB = cos(A +B)
Question 4.

Reduce each of the following expressions to the Sine And Cosine ofA single expression:

24 cos x + 7 sin x
Answer:

Let f(x) = 24 cos x + 7 sin x


Dividing and multiplying by √(242 + 72) = √625 = 25,



Sine of expression:


where sin α = 24/25 And cos α = 7/25


We know that sinA cosB + cosA sinB = sin(A +B)



Cosine of the expression:



We know that cosA cosB + sinA sinB = cos(A -B)
Question 5.

Show that Sin 1000 – Sin 100 is positive.
Answer:

Let f(x) = sin 100° – sin 10°


Dividing And multiplyingBy √(1 + 1) = √2,





We know that cosA cosB – sinA sinB = cos(A +B)



∴ f(x) = √2 cos 55°
Question 6.

Prove that lies between and .
Answer:

Let f(x) =(2√3 + 3) sin x + 2√3 cos x


HereA = 2√3 , B = 2√3 + 3 And c = 0





Hence proved.
Very Short Answer
Question 1.

If α + β - γ = π, and sin2 α + sin2 β – sin2 γ = λ sin α sin β cos γ, then write the value of λ.
Answer:

α + β = π + γ

Sin (α + β) =sin (π + γ)


Sin (α) cos (β) + sin (β) cos (α)=-sin(γ)


Take square both side


[sin(α)cos(β)+sin(β)cos(α)]2=sin2(γ)


sin2(α)cos2(β)+sin2(β)cos2(α)+2 Sin(α)cos(β)sin(β)cos(α)= sin2(γ)


sin 2(α)[1-sin2(β)]+sin2(β)[1-sin2(α)]+2 Sin(α)cos(β)sin(β)cos(α)= sin2(γ)


sin 2(α)-Sin2(α)sin2(β)+sin2(β)-sin2(β)sin2(α) -sin2(γ)=- 2Sin(α)cos(β)sin(β)cos(α)


sin 2(α)+sin2(β)-sin2(γ)=2Sin2(α)sin2(β)- 2Sin(α)cos(β)sin(β)cos(α)


sin 2(α)+sin2(β)-sin2(γ)=-2Sin(α)sin(β)[ cos(β) cos(α)- Sin(α)sin(β)]


sin 2(α)+sin2(β)-sin2(γ)=-2Sin(α)sin(β) cos(α+ β)


sin 2(α)+sin2(β)-sin2(γ)=2Sin(α)sin(β) sin(γ)
Question 2.

If then write the value of


Answer:








Question 3.

Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
Answer:

the maximum value of (a

So maximum value


=5


the minimum value of (a


minimum value= -5
Question 4.

Write the maximum values of 12 sin x – 9 sin2 x.
Answer:




f(x)ϵ-(3[-1,1]-2)2+4


f(x)ϵ-([-3,3]-2)2+4


f(x)ϵ-([-5,1])2+4


f(x)ϵ-[0,25]+4


f(x)ϵ[-25,0]+4


f(x)ϵ[-21,4]
Question 5.

If 12 sin x – 9 sin2 x attains its maximum value at x = α, then write the value of sin α.
Answer:

f(x)=12sin(x) - 9sin2(x)

f’(x)= 12cos(x)-18sin(x)cos(x)


f’(x)=0 for the maximum value of f(x)


12cos(x)-18sin(x)cos(x)=0
Question 6.

Write the interval in which the values of lie.
Answer:





f(x)ϵ[-7,7]+3


f(x)ϵ[-4,10]
Question 7.

If tan (A + B) = p and tan (A – B) = q, then write the value of tan 2B.
Answer:





Question 8.

If then write the value of tan x tan y.
Answer:

use componendo and dividendo rule
Question 9.

If then write the value of ab + bc + ca.
Answer:


a=k


b—2k


c=-2k


ab + bc + ca.=0
Question 10.

If A + B = C, then write the value of tan A tan B tan C.
Answer:

A+B=C


tan A + tan B = tan C(1- tan A tan B)


tan C – tan A – tan B = tan A tan B tan C.
Question 11.

If sin α – sin β = a and cos α + cos β = b, then write the value of cos (α + β).
Answer:

sin α – sin β = a

cos α + cos β = b


(sin α – sin β )2= a2 …..1


(cos α + cos β)2= b2 …….2


add both equations



value of



Question 12.

If and then write the value of α + β lying in the interval (0, π/2).
Answer:

Assume x=0






=1


(α+β) = tan-1 1
Mcq
Question 1.

Mark the correct alternative in the following:

The value of is

A. 1/2

B. √3/2

C. 1

D. 0
Answer:





Question 2.

Mark the correct alternative in the following:

If A + B + C = π, then sec A (cos B cos C – sin B sin C) is equal to

A. 0

B. -1

C. 1

D. None of these
Answer:

B+C=

TAKE BOTH SIDE COS


Cos (B+C)=cos(π-A)


(cos B cos C – sin B sin C) = -cos(A)


sec A (cos B cos C – sin B sin C) is equal to =-1
Question 3.

Mark the correct alternative in the following:

tan 20o + tan 40o + √3 tan 20o tan 40o is equal to

A.

B.

C.

D. 1
Answer:

tan 20o + tan 40o + √3 tan 20o tan 40o

tan 60o (1- tan 20o tan 40o)+ √3 tan 20o tan 40o
Question 4.

Mark the correct alternative in the following:

If and then the value of A + B is

A. 0

B.

C.

D.


Answer:

put a=0

tan(A)=0


tan(B)=1





A+B=
Question 5.

Mark the correct alternative in the following:

If 3 sin x + 4 cos x = 5, then 4 sin x – 3 cos x =

A. 0

B. 5

C. 1

D. None of these
Answer:

3sin(x)+4cos(x)=5


cos(37°-x)=cos 0° (∵ x=37°)


4sin(x)-3cos(x)=k
Question 6.

Mark the correct alternative in the following:

If in a ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =

A. 6

B. 1

C. 1/6

D. None of these
Answer:

A+B=π-C
Question 7.

Mark the correct alternative in the following:

tan 3A – tan 2A – tan A is equal to

A. tan 3 A tan 2 A tan A

B. - tan 3 A tan 2 A tan A

C. tan A tan 2 A – tan 2 A tan 3 A – tan 3 A tan A

D. None of these
Answer:

tan(A)+tan(B)+tan(C)=tan(A) tan(B)tan(C)

tan 3A – tan 2A – tan A= tan 3 A tan 2 A tan A
Question 8.

Mark the correct alternative in the following:

If then is equal to

A. tan A tan B tan C

B. 0

C. 1

D. None of these
Answer:

A+B=π-C
Question 9.

Mark the correct alternative in the following:

If and where P and Q both are acute angles. Then, the value of P – Q is

A.

B.

C.

D.


Answer:

,

,


Cos(p-q)=cos(p)cos(q)+sin(p)sin(q)
Question 10.

Mark the correct alternative in the following:

If cot (α + β) = 0, then sin (α + 2 β) is equal to

A. sin α

B. cos 2 β

C. cos α

D. sin 2 α
Answer:

cot (α + β) = 0


sin (α + 2 β)= sin (α +β) cos(β) + sin(β) cos (α +β)


put


sin (α + 2 β)= cos(β)
Question 11.

Mark the correct alternative in the following:

is equal to

A. tan 55o

B. cot 55o

C. - tan 35o

D. - cot 35o
Answer:





= tan 55°
Question 12.

Mark the correct alternative in the following:

The value of is

A.

B. 0

C.

D.


Answer:

cos2 A –sin2 B

=cos(A+B)cos(A-B)
Question 13.

Mark the correct alternative in the following:

If then

A.

B.

C.

D.


Answer:




Question 14.

Mark the correct alternative in the following:

If sin (π cos x) = cos (π sin x), then sin 2x =

A.

B.

C.

D. None of these
Answer:

sin(π cos x)=cos(π sin x)




Put n=0



,


,


Take square both side


,


,


Question 15.

Mark the correct alternative in the following:

If and then the value of θ + ϕ is

A.

B. π

C. 0

D.


Answer:





Question 16.

Mark the correct alternative in the following:

The value of cos (36o – A) cos (36o + A) + cos (54o + A) cos (54o – A) is

A. sin 2 A

B. cos 2A

C. cos 3A

D. sin 3A
Answer:

cos(54o + A) =sin(36o – A)

cos(54o - A) =sin(36o + A)


cos (36o–A) cos (36o+A)+ sin(36o-A)sin(36o+A)=cos(2A)
Question 17.

Mark the correct alternative in the following:

If tan (π/4 + x) + tan (π/4 – x) = a, then tan2 (π /4 + x) + tan2 (π /4 – x) =

A. a2 + 1

B. a2 + 2

C. a2 – 2

D. None of these
Answer:



= a2 - 2
Question 18.

Mark the correct alternative in the following:

If then the smallest positive value of B is

A.

B.

C.

D.


Answer:




Question 19.

Mark the correct alternative in the following:

If A – B = π /4, then (1 + tan A) (1 – tanB) is equal to

A. 2

B. 1

C. 0

D. 3
Answer:


1+tan A tan B=tan A-tan B


tan A-tan B-tan A tan B=1


add both side 1


1+tan A-tan B-tan A tan B=1+1


(1+tan A)(1+tan B)=2


Case2:


put A=0 AND


(1+tan A )(1-tan A )=1× 2
Question 20.

Mark the correct alternative in the following:

The maximum value of is

A. 1/2

B. 3/2

C. 1/4

D. 3/4
Answer:


1-


[ ]
Question 21.

Mark the correct alternative in the following:

If and tan A tan B = 2, then

A.

B.

C.

D.


Answer:

Cos(A-B) =cos(A)cos(B)+sin(A)sin(B)
Question 22.

Mark the correct alternative in the following:

If tan 69o + tan 66o – tan 69o tan 66o = 2k, then k =

A. -1

B. 1/2

C. -1/2

D. None of these
Answer:




-1+tan 69° tan66°=tan 69°+tan66°


tan 69°+tan66°-tan 69° tan66°=-1


2k=-1
Question 23.

Mark the correct alternative in the following:

If and then α + β is equal to

A.

B.

C.

D.


Answer:

put x=1





tan(α+β)=1