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Proof of the Sum of Squares of Sine and Cosine

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \cos^2\,x+sin^2\,x$	$=$	$\displaystyle $	$\displaystyle (\cos x+i\,\sin x)\,(\cos x-i\,\sin x)$	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle $	$\displaystyle (\cos\,x+i\,\sin x)\,(\cos(-x)+i\,\sin(-x))$	$\displaystyle $	$\displaystyle $	Cosine Function is Even, Sine Function is Odd
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle $	$\displaystyle e^{ix}\,e^{-ix}$	$\displaystyle $	$\displaystyle $	Euler's Formula
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle $	$\displaystyle 1$	$\displaystyle $	$\displaystyle $

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \cos^2\,x+sin^2\,x\)	\(=\)	\(\displaystyle \)	\(\displaystyle (\cos x+i\,\sin x)\,(\cos x-i\,\sin x)\)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle (\cos\,x+i\,\sin x)\,(\cos(-x)+i\,\sin(-x))\)	\(\displaystyle \)	\(\displaystyle \)	Cosine Function is Even, Sine Function is Odd
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle e^{ix}\,e^{-ix}\)	\(\displaystyle \)	\(\displaystyle \)	Euler's Formula
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle 1\)	\(\displaystyle \)	\(\displaystyle \)