# How do you verify cos^4x - sin^4x = cos^2x - sin^2x?

May 22, 2015

${\cos}^{4} x - {\sin}^{4} x$

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

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

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

based on the identities:

$\left({a}^{2} - {b}^{2}\right) = \left(a - b\right) \left(a + b\right)$

${\cos}^{2} x + {\sin}^{2} x = 1$ from Pythagoras (right angled triangle with hypotenuse of length 1).