<s>[INST] 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?
 [/INST] Both of $\sin(x),\cos(x)$ must be nonnegative since, if one of them was negative, the equation $\sin^3(x)+\cos^3(x)=1$ would imply that the other one is more than $1$, contradiction.

Thus, we have $0\le \sin(x) \le 1$ and $0\le \cos(x)\le 1$.

If both of $\sin(x),\cos(x)$ are less than $1$, then, since they are both nonnegative, we would have 
$$
\begin{cases}
0\le \sin^3(x) < \sin^2(x)\\[4pt]
0\le \cos^3(x) < \cos^2(x)\\
\end{cases}
$$
but that would imply
$$\sin^3(x)+\cos^3(x) < \sin^2(x)+\cos^2(x)=1$$
contradiction.

It follows that one of $\sin(x),\cos(x)$ must be equal to $1$, and the other must be equal to zero.

From that information, I'm sure you can finish the solution.
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