How can I find all the solutions of $\sin^5x+\cos^3x=1$ 
Find all the solutions of $$\sin^5x+\cos^3x=1$$

Trial:$x=0$ is a solution of this equation. How can I find other solutions (if any). Please help.

Use what you know about the magnitudes of $\sin x$ and $\cos x$.
$$
\begin{aligned}
\sin^5x+\cos^3x &\le |\sin^5x+\cos^3x| \\
&\le |\sin^5 x| + |\cos^3 x| \\
&\le |\sin^2 x| + |\cos^2 x| \\
&= \sin^2 x + \cos^2 x\\
&= 1
\end{aligned}
$$
The inequality where the exponents are changed is only satisfied if the individual terms are equal: $\sin x$ and $\cos x$ must both be $0$ or $1$. Put that into the original equation, and you get $\sin x = 1$ or $\cos x = 1$. So,  $x= \frac\pi2 + 2n\pi$ or $x = 2m\pi$.
