Question

prove that sin5x=5sinx-20sin^3x+16sin^5x

Answer 1

sin 5x = sin(4x + x)

= sin 4x cos x + cos 4x sin x

= 2 sin 2x cos 2x cos x + (1 - 2 sin2 2x) sin x

= 4 sin x cos x cos 2x cos x + sin x - 2 sin^2 2x sin x

= 4 sin x cos^2 x (1 - 2 sin^2 x) + sin x - 2 (2 sin x cos x)^2 sin x

= 4 sin x (1 - sin^2 x) (1 - 2 sin^2 x) + sin x - 2 (4 sin^2 x cos^2 x) sin x

= 4 sin x (1 - 3 sin^2 x + 2 sin^4 x) + sin x - 2 [4 sin^3 x * (1 - sin^2 x)]

= 4 sin x - 12 sin^3 x + 8 sin^5 x + sin x - 8 sin^3 x + 8 sin^5 x

= 5sinx-20sin^3 x+16sin^5 x