<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] This is what I tried after reading your answers, please correct me if there are mistakes!$$cos^3+sin^3=1$$ $$cos^3+sin^3=cos² +sin²$$ $$cos^3-cos²+sin^3-sin²=0$$ $$cos²(cos-1) + sin²(sin-1)=0$$ $$(1-sin²)(cos-1)+(1-cos²)(sin-1)=0$$ $$ (1+sin)(1-sin)(cos-1)+(1+cos)(1-cos)(sin-1)=0 $$ $$(1+sin)(1-sin)(cos-1)-(cos-1)(1-cos)(sin-1)=0$$ $$(cos-1)[ (1+sin)(1-sin)-(sin-1)(1-cos)]=0$$ $$(cos-1)[ (1+sin)(1-sin)+(1-sin)(1-cos)]=0$$ $$(cos-1)[ (1-sin)(1+sin)(1-cos)]=0$$ $$(cos-1)(1-sin²)(1-cos)=0$$ $$(1-sin²)(1-cos)=0$$ $$1-sin²=0$$ $$sin²=1$$ $$sin=1$$
or: $$1-cos=0$$ $$cos=1$$
If: $sin=1$ ⇒  $sin^3=1$
Then:$$cos^3+sin^3=1$$ $$cos^3+1=1$$ $$cos^3=0$$ $$cos=0$$
If: $cos=1$ ⇒  $cos^3=1$
Then:$$cos^3+sin^3=1$$ $$sin^3+1=1$$ $$sin^3=0$$ $$sin=0$$
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