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

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

Answered by john_santu last updated on 14/May/21

$$\sqrt{\sin x}⩾0$$$$\Rightarrow \sqrt{\sin x}\cos x(1-\sin {}^{4}x-\cos {}^{2}x)=0$$$$\left(1\right)\sqrt{\sin x}=0\to x=2n\pi$$$$\left(2\right)\cos x=0\to x=\pm \frac{\pi }{2}+2n\pi$$$$\left(3\right)1-{(1-\cos {}^{2}x)}^2-\cos {}^{2}x=0$$$$\Rightarrow (1-\cos {}^{2}x)(1-(1-\cos {}^{2}x))=0$$$$\Rightarrow (1-\cos {}^{2}x)\cos {}^{2}x=0$$$$\Rightarrow \cos x=1,\to x=2n\pi$$$$\Rightarrow \cos x=-1,\to x=\pi +2n\pi$$

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