# How do you simplify sin^4(x)-cos^4(x)-sin^2(x)+cos^2(x) ?

Oct 21, 2016

${\sin}^{4} \left(x\right) - {\cos}^{4} \left(x\right) - {\sin}^{2} x + {\cos}^{2} \left(x\right) = 0$

#### Explanation:

Use:

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

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

as follows:

${\sin}^{4} \left(x\right) - {\cos}^{4} \left(x\right) - {\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right)$

$= \left({\sin}^{4} \left(x\right) - {\cos}^{4} \left(x\right)\right) - \left({\sin}^{2} \left(x\right) - {\cos}^{2} \left(x\right)\right)$

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

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

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

$= 0$