Difference of Fourth Powers of Cosine and Sine

Theorem

$\sin^4 x - \cos^4 x = \sin^2 x - \cos^2 x$

where $\sin$ and $\cos$ denote sine and cosine respectively.

$\ds \sin^4 x - \cos^4 x$

$=$

$\ds \sin^2 x \left({1 - \cos^2 x}\right) - \cos^2 x \left({1 - \sin^2 x}\right)$

$\ds $

$=$

$\ds \sin^2 x - \sin^2 x \ \cos^2 x - \cos^2 x + \sin^2 x \ \cos^2 x$

$\ds $

$=$

$\ds \sin^2 x - \cos^2 x$

$\blacksquare$

$\ds \sin^2 x - \cos^2 x$

$=$

$\ds \left({\sin^2 x - \cos^2 x}\right) \left({\sin^2 x + \cos^2 x}\right)$

$\ds $

$=$

$\ds \sin^4 x - \cos^4 x$

$\blacksquare$