Question

Prove that: $sin^{2} 6x - sin^{2}4x=sin 2x sin 10x$.

Answer 1

LHS = sin² 6x - sin² 4x

= (sin6x - sin4x)(sin6x + sin4x) [ from algebraic identity , a² - b² = (a - b)(a + b)]

now use formula,
sinC - sinD = 2cos(C + D)/2.sin(C - D)/2
sinC + sinD = 2sin(C + D)/2.cos(C - D)/2.

so, sin6x - sin4x = 2cos(6x + 4x)/2.sin(6x -4x)/2
= 2cos5x.sinx

sin6x + sin4x = 2sin(6x + 4x)/2.cos(6x - 4x)/2
=2sin5x.cosx

(sin6x - sin4x)(sin6x + sin4x) = (2cos5x.sinx)(2sin5x.cosx)

= (2sin5x.cos5x)(2sinx.cosx)

use formula,
2sinA.cosA = sin2A.

= sin10x.sin2x = RHS