Question

Prove that cot 4x (sin 5x + sin 3x) = cot x(sin 5x - sin 3x).

Answer 1

LHS = cot 4x (sin 5x + sin 3x)

`= cot 4x xx 2 sin ((5x + 3x)/2) cos ((5x - 3x)/2)`

`= 2 (cos 4x)/(sin 4x) xx sin 4x xx cos x` = 2 cos 4x cos x

RHS = cot x (sin 5x - sin 3x)

`= cot x xx 2 sin ((5x - 3x)/2) * cos ((5x + 3x)/2)`

`[sin "A" - sin "B" = 2 sin (("A - B")/2) cos(("A + B")/2)]`

`= (cos x)/cancel(sin x) xx 2 cancel (sin x) cos 4x`

= 2 cos x cos 4x

LHS = RHS.

Hence proved.