Solve $\sin^{3}x+\cos^{3}x=1$ 
Solve for $x\\ \sin^{3}x+\cos^{3}x=1$

$\sin^{3}x+\cos^{3}x=1\\(\sin x+\cos x)(\sin^{2}x-\sin x\cdot\cos x+\cos^{2}x)=1\\(\sin x+\cos x)(1-\sin x\cdot\cos x)=1$
What should I do next?

As an alternative by
$$\sin^3 x+ \cos ^3 x=1 \iff \sin x\cdot \sin^2+\cos x\cdot \cos^2 x=1$$
since $\sin^2 x+ \cos ^2 x=1$, the given equality is a weighted mean of $\sin x$ and $\cos x$ which holds if and only if 


*

*$\sin x=1,\,\cos x=0$
or


*

*$\cos x=1,\,\sin x=0$
