Sum of Squares of Sine and Cosine/Proof 4
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Theorem
- $\cos^2 x + \sin^2 x = 1$
Proof
\(\ds \cos^2 x + \sin^2 x\) | \(=\) | \(\ds \left({\cos x + i \, \sin x}\right) \, \left({\cos x - i \, \sin x}\right)\) | factoring over the complex numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \left({\cos x + i \, \sin x}\right) \, \left({\cos \left({-x}\right) + i \, \sin \left({-x}\right)}\right)\) | Cosine Function is Even and Sine Function is Odd | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{ix} \, e^{-ix}\) | Euler's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
$\blacksquare$