RD Sharma Class 10 Solutions Chapter 6 Trigonometric Identities

RD Sharma Class 10 Solutions Trigonometric Identities Exercise 6.1

Prove the following trigonometric identities :
Question 1.
(1 – cos² A) cosec² A = 1
Solution:
(1 – cos² A) cosec² A = 1
L.H.S. = (1 – cos² A) cosec² A = sin² A cosec² A  (∵ 1 – cos² A = sin² A)
= (sin A cosec A)² = (l)² = 1 = R.H.S.  {sin A cosec A = 1 }

Question 2.
(1 + cot² A) sin² A = 1
Solution:
(1 + cot² A) sin² A = 1
L.H.S. = (1 + cot² A) (sin² A)
= cosec² A sin² A {1 + cot² A = cosec² A}
= [cosec A sin A]²
= (1)²= 1 = R.H.S. (∵ sin A cosec A = 1

Question 3.
tan² θ cos²  θ = 1- cos²  θ
Solution:

Question 4.

Solution:

Question 5.
(sec² θ – 1) (cosec² θ – 1) = 1
Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.

Solution:

Question 11.

Solution:

Question 12.

Solution:

Question 13.

Solution:

Question 14.

Solution:

Question 15.

Solution:

Question 16.
tan²  θ – sin²  θ = tan²  θ sin² θ
Solution:

Question 17.
(sec θ + cos θ ) (sec θ – cos θ ) = tan²  θ + sin² θ
Solution:

Question 18.
(cosec θ + sin θ) (cosec θ – sin θ) = cot² θ + cos² θ
Solution:

Question 19.
sec A (1 – sin A) (sec A + tan A) = 1 (C.B.S.E. 1993)
Solution:

Question 20.
(cosec A – sin A) (sec A – cos A) (tan A + cot A) = 1
Solution:

Question 21.
(1 + tan²  θ) (1 – sin θ) (1 + sin θ) = 1
Solution:

Question 22.
sin² A cot² A + cos² A tan² A = 1 (C.B.S.E. 1992C)
Solution:

Question 23.

Solution:

Question 24.

Solution:

Question 25.

Solution:

Question 26.

Solution:

Question 27.

Solution:

Question 28.

Solution:

Question 29.

Solution:

Question 30.

Solution:

Question 31.
sec⁶ θ= tan⁶  θ + 3 tan²  θ sec²  θ + 1
Solution:

Question 32.
cosec⁶  θ = cot⁶  θ+ 3cot²θ cosec²  θ + 1
Solution:

Question 33.

Solution:

Question 34.

Solution:

Question 35.

Solution:

Question 36.

Solution:

Question 37.

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Question 38.

Solution:

Question 39.

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Question 40.

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Question 41.

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Question 42.

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Question 43.

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Question 44.

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Question 45.

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Question 46.

Solution:

Question 47.


Solution:

Question 48.

Solution:

Question 49.
tan² A + cot² A = sec² A cosec² A – 2
Solution:

Question 50.

Solution:

Question 51.

Solution:

Question 52.

Solution:

Question 53.

Solution:

Question 54.
sin² A cos² B – cos² A sin² B = sin² A – sin² B.
Solution:
L.H.S. = sin² A cos² B – cos² A sin² B
= sin² A (1 – sin² B) – (1 – sin² A) sin² B
= sin² A – sin² A sin² B – sin² B + sin² A sin² B
= sin² A – sin² B
Hence, L.H.S. = R.H.S.

Question 55.

Solution:

Question 56.
cot² A cosec² B – cot² B cosec² A = cot² A – cot² B
Solution:

Question 57.
tan² A sec² B – sec² A tan² B = tan² A – tan² B
Solution:

Prove the following identities: (58-75)
Question 58.
If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that x¹ – y² = a² – b¹. [C.B.S.E. 2001, 20O2C]
Solution:
x – a sec θ + b tan θ
y = a tan θ + b sec θ
Squaring and subtracting, we get
x²-y² = {a sec θ + b tan θ)² – (a tan θ + b sec θ)²
= (a² sec² θ + b² tan²  θ + 2ab sec θ x tan θ) – (a² tan²  θ + b² sec²  θ + 2ab tan θ sec θ)
= a² sec² θ + b tan²  θ + lab tan θ sec θ – a² tan²  θ – b² sec²  θ – 2ab sec θ tan θ
= a² (sec² θ – tan²  θ) + b² (tan²  θ – sec²  θ)
= a² (sec² θ – tan²  θ) – b² (sec²  θ – tan²  θ)
=  a² x 1-b² x 1 =a²-b² = R.H.S.

Question 59.
If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ±3
Solutioon:

Question 60.
If cosec θ + cot θ = mand cosec θ – cot θ = n,prove that mn= 1
Solution:

Question 61.

Solution:

Question 62.

Solution:

Question 63.

Solution:

Question 64.

Solution:

Question 65.

Solution:

Question 66.
(sec A + tan A – 1) (sec A – tan A + 1) = 2 tan A
Solution:

Question 67.
(1 + cot A – cosec A) (1 + tan A + sec A) = 2
Solution:

Question 68.
(cosec θ – sec θ) (cot θ – tan θ) = (cosec θ + sec θ) (sec θ cosec θ-2)
Solution:

Question 69.
(sec A – cosec A) (1 + tan A + cot A) = tan A sec A – cot A cosec A
Solution:

Question 70.

Solution:

Question 71.

Solution:

Question 72.

Solution:

Question 73.
sec⁴ A (1 – sin⁴ A) – 2tan² A = 1
Solution:

Question 74.

Solution:

Question 75.

Solution:

Question 76.

Solution:

Question 77.
If cosec θ – sin θ = a³, sec θ – cos θ = b³, prove that a²b² (a² + b²) = 1
Solution:

Question 78.

Solution:

Question 79.

Solution:

Question 80.
If a cos θ + b sin θ = m and a sin θ – b cos θ = n, prove that a² + b² = m² + n² 
Solution:

Question 81.
If cos A + cos² A = 1, prove that sin² A + sin⁴ A = 1
Solution:
cos A + cos² A = 1
⇒ cos A = 1 – cos² A
⇒cos A = sin² A
Now, sin² A + sin⁴ A = sin² A + (sin² A)²
= cos A + cos² A = 1 = R.H.S.

Question 82.
If cos θ + cos²  θ = 1, prove that
sin¹² θ + 3 sin¹⁰  θ + 3 sin⁸  θ + sin⁶  θ + 2 sin⁴  θ + 2 sin²  θ-2 = 1
Solution:

Question 83.
Given that :
(1 + cos α) (1 + cos β) (1 + cos γ) = (1 – cos α) (l – cos β) (1 – cos γ)
Show that one of the values of each member of this equality is sin α sin β sin γ
Solution:

Question 84.

Solution:

Question 85.

Solution:

Question 86.
If sin θ + 2cos θ = 1 prove that 2sin θ – cos θ = 2. [NCERT Exemplar]
Solution:

RD Sharma Class 10 Solutions Chapter 6 Exercise 6.2

RD Sharma Class 10 Solutions Trigonometric Identities Exercise 6.2

Question 1.
If cos θ = , find all other trigonometric ratios of angle θ
Solution:


Question 2.
If sin θ =  , find all other trigono­metric ratios of angle θ
Solution:

Question 3.
If tan θ = , find the value of 
Solution:
<

Question 4.
If tan θ = , find the value of 
(C.B.S.E. 1995)
Solution:

Question 5.

Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.
If tan θ = 3sin θ , find the value of sin² θ  – cos² θ .          (C.B.S.E. 2001)
Solution:

Question 11.

Solution:

Question 12.
If sin θ + cos θ = cos (90° – θ), find cot θ
Solution:

Question 13.
If 2sin² θ – cos² θ = 2, then find the value of θ. [NCERT Exemplar]
Solution:

Question 14.
If tan θ-1=0, find the value of sin² – cos² θ. [NCERT Exemplar]
Solution: