Question:
Proove that sin5x = 5sinx - 20sin^3x 16sin^5x
Answer:
Given, 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 sin2 2x sin x
= 4 sin x cos2 x (1 - 2 sin2 x) + sin x - 2 (2 sin x cos x)2 sin x
= 4 sin x (1 - sin2 x) (1 - 2 sin2 x) + sin x - 2 (4 sin2 x cos2 x) sin x
= 4 sin x (1 - 3 sin2 x + 2 sin4 x) + sin x - 2 [4 sin3 x * (1 - sin2 x)]
= 4 sin x - 12 sin3 x + 8 sin5 x + sin x - 8 sin3 x + 8 sin5 x
= 16 sin5 x - 20 sin3 x + 5 sinx
So, sin 5x = 16 sin5 x - 20 sin3 x + 5 sinx
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