Question

If x = 30°, then prove that

(i) sin 3x = 3 sin x – 4 sin³ x

(ii) tan2x = 2tanx/(1 - tan²x)

(iii) sinx = √((1 - cos2x)/2)

(iv) cos 3x = 4 cos³ x – 3 cos x

Answer 1

(i) sin 3x = 3 sin x – 4 sin³ x

L.H.S. = sin 3x

= sin 3 × 30° [∵ Given x = 30°]

= sin 90°

= 1
R.H.S. = 3 sin x – 4 sin³ x
= 3sin 30° – 4 sin³ 30°

Thus, L.H.S. = R.H.S.

(ii) tan2x = 2tanx/(1 - tan²x)

L.H.S. = tan 2x

= tan 2 × 30° [∵ Given x = 30°]

= tan 60° = √3

R.H.S. = 2tanx/(1 - tan²x)

∴ L.H.S. = R.H.S.

(iii) sinx = √((1 - cos2x)/2)

∴ L.H.S. = R.H.S.

(iv) cos 3x = 4 cos³ x – 3 cos x

L.H.S. = cos 3x = cos 3 × 30°

= cos 90°

= 0 [∵ Given x = 30°]

R.H.S. = 4 cos³ x – 3 cos x 

= 4 cos³ 30° – 3 cos 30°

L.H.S. = R.H.S.