# How do you prove cos^4x - sin^4x = 1 - 2sin^2x?

Apr 18, 2016

Start by factoring the left side as a difference of squares.

#### Explanation:

${\cos}^{4} x - {\sin}^{4} x = 1 - 2 {\sin}^{2} x$

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

Now, applying the pythagorean identity ${\cos}^{2} x + {\sin}^{2} x = 1$:

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

Rearranging the previously stated pythagorean identity to solve for sin:

${\cos}^{2} x + {\sin}^{2} x = 1$

${\cos}^{2} x = 1 - {\sin}^{2} x$

Substituting:

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

$1 - 2 {\sin}^{2} x = 1 - 2 {\sin}^{2} x \to$ identity proved

Hopefully this helps!