Sine and cosine/R/Properties of periodicity/Fact/Proof

Proof

Due to the circle equation

${}(\cos z)^{2}+(\sin z)^{2}=1\,,$

we have ${}{\left(\sin {\frac {\pi }{2}}\right)}^{2}=1$, hence ${}\sin {\frac {\pi }{2}}=1$, because of the reasoning in the proof of fact. From that we deduce, with the help of the addition theorems, the relations between sine and cosine as mentioned in (3), e.g.

${}\cos {\left(z+{\frac {\pi }{2}}\right)}=\cos z\cos {\left({\frac {\pi }{2}}\right)}-\sin z\sin {\left({\frac {\pi }{2}}\right)}=-\sin z\,.$

Hence it is enough to prove the statements for cosine. All statements follow from the definition of ${}\pi$ and from (3).

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