buktikan identitas trigonometri berikuta)16 sin^5=10 sina-5sin 3a+sin 5a

Trigonometry. Level: EXPERT!!!

First expand sin 5x.
sin 5a = sin (3a + 2a)
= sin 3a cos 2a + cos 3a sin 2a
= (3 sin a - 4 sin³ a)(2 cos² a - 1) + (4 cos³ a - 3 cos a)(2 sin a cos a)
= (3 sin a - 4 sin³ a)[2 (1 - sin² a) - 1] + (8 sin a cos⁴ a - 6 sin a cos² a)
= (3 sin a - 4 sin³ a)(1 - 2 sin² a) + [8 sin a (1 - sin² a)² - 6 sin a (1 - sin² a)]
= 5 sin a - 20 sin³ a + 16 sin⁵ a

Then we proving from right side to left side.
10 sin a - 5 sin 3a + sin 5a
= 10 sin a - 5 (3 sin a - 4 sin³ a) + 5 sin a - 20 sin³ a + 16 sin⁵ a
= 16 sin⁵ a

So, proved that 16 sin⁵ a = 10 sin a - 5 sin 3a + sin 5a