Difference of Fourth Powers of Cosine and Sine/Proof 1
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Theorem
- $\sin^4 x - \cos^4 x = \sin^2 x - \cos^2 x$
Proof
\(\ds \sin^4 x - \cos^4 x\) | \(=\) | \(\ds \sin^2 x \left({1 - \cos^2 x}\right) - \cos^2 x \left({1 - \sin^2 x}\right)\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin^2 x - \sin^2 x \ \cos^2 x - \cos^2 x + \sin^2 x \ \cos^2 x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin^2 x - \cos^2 x\) |
$\blacksquare$