Sum of Squares of Sine and Cosine/Proof 4

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Theorem

$\cos^2 x + \sin^2 x = 1$


Proof

\(\displaystyle \cos^2 x + \sin^2 x\) \(=\) \(\displaystyle \left({\cos x + i \, \sin x}\right) \, \left({\cos x - i \, \sin x}\right)\) factoring over the complex numbers
\(\displaystyle \) \(=\) \(\displaystyle \left({\cos x + i \, \sin x}\right) \, \left({\cos \left({-x}\right) + i \, \sin \left({-x}\right)}\right)\) Cosine Function is Even and Sine Function is Odd
\(\displaystyle \) \(=\) \(\displaystyle e^{ix} \, e^{-ix}\) Euler's Formula
\(\displaystyle \) \(=\) \(\displaystyle 1\)

$\blacksquare$