Question

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

Answer 1

It is known that

$sin A+sin B=2sin(\frac{A+B}{2}).cos(\frac{A-B}{2})$

L.H.S. = sin x+ sin 3x+sin 5x+sin 7x

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

$=2sin(\frac{x+5x}{2}).cos(\frac{x-5x}{2})+2\,sin(\frac{3x+zx}{2})cos(\frac{3x-7x}{2})$

=2 sin 3x cos(-2x)+2sin5x cos(-2x)

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

=2 cos 2x[sin3x+sin5x]

$=2cos2x[2sin(\frac{3x+5x}{2}).cos(\frac{3x-5x}{2})]$

=2 cos 2x [2 sin 4x.cos(-x)]

=4 cos 2x sin 4x cos x=R.H.S.