Question

How do you simplify ${\cos }^{4}x-{\sin }^{4}x$?

Answer 1

Simplifying the expression using trigonometric identities:

$$\begin{gathered} {\cos }^{4}x-{\sin }^{4}x=\left({\cos }^{2}x-{\sin }^{2}x\right)\left({\cos }^{2}x+{\sin }^{2}x\right) \\ =\cos 2x({\sin }^{2}x+{\cos }^{2}x=1and{\cos }^{2}x-{\sin }^{2}x=\cos 2x) \end{gathered}$$

Therefore, ${\cos }^{4}x-{\sin }^{4}x=\cos 2x$.