ProofWiki:Sandbox
From ProofWiki
| Welcome to the Sandbox! Play safe and have fun!
Use Proofwiki:Sandbox/Template for experimental template Warning: The sandbox is not permanent. EXPECT ANYTHING YOU PUT HERE TO BE DELETED! |
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 | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\displaystyle 1\) | \(\displaystyle \) | \(\displaystyle \) |
$\blacksquare$