Sum of Squares of Sine and Cosine/Proof 4

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

\(\displaystyle \)

\(=\)

\(\displaystyle 1\)

$\blacksquare$