Question

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

Answer 1

It is known that $sinA+sinB=2sin(\frac{A+B}{2})cos(\frac{A-B}{2}),\,sinA-sinB=2cos(\frac{A+B}{2})sin(\frac{A-B}{2})$

∴L.H.S. = sin26x-sin24x

= (sin 6x+sin 4x) (sin 6x-sin 4x)

$=[2\,sin(\frac{6x+4x}{2})cos(\frac{6x-4x}{2})][2\,cos(\frac{6x+4x}{2}).sin(\frac{6x-4x}{2})]$

= (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