Difference of Fourth Powers of Cosine and Sine/Proof 1

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

\(\ds \)

\(=\)

\(\ds \sin^2 x - \sin^2 x \ \cos^2 x - \cos^2 x + \sin^2 x \ \cos^2 x\)

\(\ds \)

\(=\)

\(\ds \sin^2 x - \cos^2 x\)

$\blacksquare$