## Trigonometry (11th Edition) Clone

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