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?

Let use by $t= \tan (x/2)$


*

*$\sin x=\frac{2t}{1+t^2}$

*$\cos x=\frac{1-t^2}{1+t^2}$
to obtain 
$$2t^6-8t^3+6t^2=0$$
$$\iff t^2(t^4-4t+3)=t^2(t-1)^2(t^2+2t+3)=0$$
and since $t^2+2t+3>0$ the solutions are


*

*$t=0 \implies \frac x 2=k\pi\implies x=2k\pi$

*$t=1 \implies \frac x 2=\frac{\pi}4+k\pi \implies x=\frac{\pi}2+2k\pi $
