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Find all the solutions of the equation $\sin^5(x) + \cos^3(x) =1.$ I tried it and get $(1-\cos(x))(\sin^3(x)-\cos^3(x) -\cos^2(x) -1)=0$ but I have no idea how to proceed from here. please help me to solve it. Thanks in advance.

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HINT:

$$\sin^5x\le\sin^2x,\cos^3x\le\cos^2x\implies\sin^5x+\cos^3x\le\sin^2x+\cos^2x=1$$

The equality occurs if $\sin^5x=\sin^2x$ and $\cos^3x=\cos^2x$

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