sin-x-cos-x-sin-5-x-cos-x-cos-3-x-sin-x-

Question Number 140907 by bramlexs22 last updated on 14/May/21

\sqrt{\sin x} \cos x - \sqrt{\sin^{5} x} \cos x = \cos^{3} x \sqrt{\sin x}

Answered by john_santu last updated on 14/May/21

\sqrt{\sin x} ⩾ 0

\Rightarrow \sqrt{\sin x} \cos x (1 - \sin^{4} x - \cos^{2} x) = 0

(1) \sqrt{\sin x} = 0 \to x = 2 n π

(2) \cos x = 0 \to x = \pm \frac{π}{2} + 2 n π

(3) 1 - {(1 - \cos^{2} x)}^{2} - \cos^{2} x = 0

\Rightarrow (1 - \cos^{2} x) (1 - (1 - \cos^{2} x)) = 0

\Rightarrow (1 - \cos^{2} x) \cos^{2} x = 0

\Rightarrow \cos x = 1, \to x = 2 n π

\Rightarrow \cos x = - 1, \to x = π + 2 n π