<s>[INST] How do I prove that if: $\cos^3(x) + \sin^3(x) = 1$ then: $\cos(x) = 0 ; \sin(x)=1$ or $\cos(x)=1 ; \sin(x)=0$ How do I prove that if:
$$\cos^3(x) + \sin^3(x) = 1$$
then:
$$\cos(x) = 0 ; \sin(x)=1 \text{ or } \cos(x)=1 ; \sin(x)=0?$$
Starting from the first expression, I couldn't figure out how to reach the conclusion. I replaced 1 by $\cos^2(x) + \sin^2(x) $ hoping to factor it but to no avail.
 [/INST] Rather than carry the notation we write $s=\sin(x)$ and $c=\cos(x)$. Subtracting the equation from $s^2+c^2=1$ yields $s^3-s^2 + c^3-c^2=0$ or $s(s^2-1)+c(c^2-1) = s(-c^2) + c(-s^2) = 0$ and so either $s=0$ or $c=0$.
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