Prove that: sin 3x + sin 2x - sin x = 4sin x cos x/2 cos 3x/2

Solution:

LHS = sin 3x + sin 2x - sin x

= sin 3x + (sin 2x - sin x)

= sin 2(3x/2) + [2cos {(2x + x) / 2} sin {(2x - x) / 2}] [Because sin A - sin B = 2cos {(A + B) / 2} sin {(A - B) / 2}]

= 2sin 3x/2 cos 3x/2 + 2cos 3x/2 sin x/2 [By double angle formulas, sin 2A = 2sin A cos A]

= 2cos (3x/2) [sin 3x/2 + sin x/2]

= 2cos (3x/2) [2sin {(3x/2 + x/2) / 2} cos {(3x/2 - x/2) / 2}] [Because sin A + sin B = 2sin {(A + B) / 2} cos {(A - B) / 2}]

= 2cos (3x/2) [2sin x cos x/2]

= 4sin x cos x/2 cos 3x/2

= RHS

NCERT Solutions Class 11 Maths Chapter 3 Exercise ME Question 7

Prove that: sin 3x + sin 2x - sin x = 4sin x cos x/2 cos 3x/2

Summary:

We got, sin 3x + sin 2x - sin x = 4sin x cos x/2 cos 3x/2. Hence Proved