## NCERT Solutions for Class 11 Maths Chapter 3 – Trigonometric Functions Ex 3.3

L.H.S. =

Prove that

L.H.S. =

Prove that

L.H.S. =

Prove that

L.H.S =

#### Question 5:

Find the value of:
(i) sin 75°
(ii) tan 15°

(i) sin 75° = sin (45° + 30°)
= sin 45° cos 30° + cos 45° sin 30°
[sin (x + y) = sin x cos y + cos x sin y]
(ii) tan 15° = tan (45° – 30°)

Prove that:

#### Question 7:

Prove that:

It is known that
∴L.H.S. =

Prove that

L.H.S. =

#### Question 10:

Prove that sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x

L.H.S. = sin (n + 1)x sin(n + 2)x + cos (n + 1)x cos(+ 2)x

#### Question 11:

Prove that

It is known that.
∴L.H.S. =

#### Question 12:

Prove that sin2 6x – sin2 4x = sin 2x sin 10x

It is known that
∴L.H.S. = sin26x – sin24x
= (sin 6x + sin 4x) (sin 6x – sin 4x)
= (2 sin 5x cos x) (2 cos 5x sin x)
= (2 sin 5x cos 5x) (2 sin x cos x)
= sin 10x sin 2x
= R.H.S.

#### Question 13:

Prove that cos2 2x – cos2 6x = sin 4sin 8x

It is known that
∴L.H.S. = cos2 2x – cos2 6x
= (cos 2x + cos 6x) (cos 2– 6x)
= [2 cos 4x cos 2x] [–2 sin 4(–sin 2x)]
= (2 sin 4x cos 4x) (2 sin 2x cos 2x)
= sin 8x sin 4x
= R.H.S.

#### Question 14:

Prove that sin 2x + 2sin 4x + sin 6x = 4cos2 x sin 4x

L.H.S. = sin 2x + 2 sin 4x + sin 6x
= [sin 2x + sin 6x] + 2 sin 4x
= 2 sin 4x cos (â€“ 2x) + 2 sin 4x
= 2 sin 4x cos 2x + 2 sin 4x
= 2 sin 4x (cos 2x + 1)
= 2 sin 4x (2 cos2 x â€“ 1 + 1)
= 2 sin 4x (2 cos2 x)
= 4cos2 x sin 4x
= R.H.S.

#### Question 15:

Prove that cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)

L.H.S = cot 4x (sin 5x + sin 3x)
= 2 cos 4x cos x
R.H.S. = cot x (sin 5x â€“ sin 3x)
= 2 cos 4x. cos x
L.H.S. = R.H.S.

Prove that

It is known that
∴L.H.S =

Prove that

It is known that
∴L.H.S. =

Prove that

It is known that
∴L.H.S. =

Prove that

It is known that
∴L.H.S. =

Prove that

It is known that
∴L.H.S. =

Prove that

L.H.S. =

#### Question 22:

Prove that cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1

L.H.S. = cot x cot 2x – cot 2x cot 3x – cot 3x cot x
= cot x cot 2x – cot 3x (cot 2x + cot x)
= cot x cot 2x – cot (2x) (cot 2x + cot x)
= cot x cot 2– (cot 2cot x – 1)
= 1 = R.H.S.

#### Question 23:

Prove that

It is known that.
∴L.H.S. = tan 4x = tan 2(2x)

#### Question 24:

Prove that cos 4x = 1 – 8sincosx

L.H.S. = cos 4x
= cos 2(2x)
= 1 – 2 sin2 2x [cos 2A = 1 – 2 sin2 A]
= 1 – 2(2 sin x cos x)2 [sin2A = 2sin A cosA]
= 1 – 8 sin2x cos2x
= R.H.S.

#### Question 25:

Prove that: cos 6x = 32 cos6 x – 48 cos4 x + 18 cos2 – 1