RD Sharma Class 10 Solutions Chapter 6 Trigonometric Identities

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 cos θ = 1- cos 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.
tan θ – sin θ = tan θ sin2 θ
Solution:

Question 17.
(sec θ + cos θ ) (sec θ – cos θ ) = tan 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 + tan 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= tan θ + 3 tan θ sec θ + 1
Solution:

Question 32.
cosec θ = cot θ+ 3cot2θ cosec θ + 1
Solution:

Question 33.

Solution:

Question 34.

Solution:

Question 35.

Solution:

Question 36.

Solution:

Question 37.

Solution:

Question 38.

Solution:

Question 39.

Solution:

Question 40.

Solution:

Question 41.

Solution:

Question 42.

Solution:

Question 43.

Solution:

Question 44.

Solution:

Question 45.

Solution:

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 tan θ + 2ab sec θ x tan θ) – (a2 tan θ + b2 sec θ + 2ab tan θ sec θ)
= a2 sec2 θ + b tan θ + lab tan θ sec θ – a2 tan θ – b2 sec θ – 2ab sec θ tan θ
= a2 (sec2 θ – tan θ) + b2 (tan θ – sec θ)
= a2 (sec2 θ – tan θ) – b2 (sec θ – tan θ)
=  ax 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 + cos θ = 1, prove that
sin12 θ + 3 sin10  θ + 3 sin θ + sin θ + 2 sin θ + 2 sin 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 = $\frac { 4 }{ 5 }$, find all other trigonometric ratios of angle θ
Solution:

Question 2.
If sin θ = $\frac { 1 }{ \sqrt { 2 } }$ , find all other trigono­metric ratios of angle θ
Solution:

Question 3.
If tan θ = $\frac { 1 }{ \sqrt { 2 } }$, find the value of $\frac { { cosec }^{ 2 }-{ sec }^{ 2 } }{ { cosec }^{ 2 }+cot^{ 2 } }$
Solution:
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Question 4.
If tan θ = $\frac { 3 }{ 4 }$, find the value of $\frac { 1-cosheta }{ 1+cosheta }$
(C.B.S.E. 1995)
Solution:

Question 5.

Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.
If $\sqrt { 3 }$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 = $\sqrt { 2 }$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 $\sqrt { 3 }$tan θ-1=0, find the value of sin2 – cos2 θ. [NCERT Exemplar]
Solution:

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