Prove that: sin x + sin 3x + sin 5x + sin 7x = 4cos x cos 2x sin 4x

Solution:

LHS = sin x + sin 3x + sin 5x + sin 7x

= (sin 5x + sin x) + (sin 7x + sin 3x)

We have  sin A + sin B = 2sin {(A + B) / 2} cos {(A - B) / 2}. So we get

= [2sin {(5x + x) / 2} cos {(5x - x) / 2}] + [2sin {(7x + 3x) / 2} cos {(7x - 3x) / 2}]

= 2sin 3x cos 2x + 2sin 5x cos 2x

= 2cos 2x (sin 5x + sin 3x)

= 2cos 2x[2sin {(5x + 3x) / 2} cos {(5x - 3x) / 2}] [Because sin A + sin B = 2sin {(A + B) / 2} cos {(A - B) / 2}]

= 2 cos2x[2sin 4x cos x]

= 4cos x cos 2x sin 4x

= RHS

NCERT Solutions Class 11 Maths Chapter 3 Exercise ME Question 5

Prove that: sin x + sin 3x + sin 5x + sin 7x = 4cos x cos 2x sin 4x

Summary:

We got, sin x + sin 3x + sin 5x + sin 7x = 4cos x cos 2x sin 4x. Hence Proved.