Answer
$\cos^4{x}-\sin^4{x}=\cos{(2x)}$
Work Step by Step
Use a graphing utility to graph the given expression. (Refer to the graph below,)
Notice that the graph is identical with the graph of $\cos{2x}$.
This means that $\cos^4{x}-\sin^4{x}=\cos{(2x)}$.
The given expression can be written as:
$(\cos^2{x})^2-(\sin^2{x})^2$
Factor the expression using the formula $a^2-b^2=(a-b)(a+b)$ where $a=\cos^2{x}$ and $b=\sin^2{x}$ to obtain:
$$(\cos^2{x})^2-(\sin^2{x})^2=\left(\cos^2{x}-\sin^2{x}\right)\left(\cos^2{x}+\sin^2{x}\right)$$
Since $\cos^2{x}+\sin^2{x}=1$. then the equation above simplifies to:
\begin{align*}
(\cos^2{x})^2-(\sin^2{x})^2&=(\cos^2{x}-\sin^2{x})(1)\\
(\cos^2{x})^2-(\sin^2{x})^2&=(\cos^2{x}-\sin^2{x})\\
\end{align*}
Recall:
$\cos{(2x)}=\cos^2{x} - \sin^2{x}$
Thus, the equation above becomes:
$$(\cos^2{x})^2-(\sin^2{x})^2=\cos{(2x)}$$
Therefore,
$$\cos^4{x}-\sin^4{x}=\cos{(2x)}$$
