Chapter 7 - Section 7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas - 7.3 Exercises - Page 562: 76

$\cos^4 x-\sin ^4 x=\cos 2x$

Start from the left side: $\cos^4 x-\sin ^4 x$ Rewrite the expression: $=(\cos^2 x)^2-(\sin ^2x)^2$ Factor it as a difference of perfect squares: $=(\cos^2 x+\sin^2x)(\cos^2 x-\sin^2x)$ Use the identities $\cos^2 x+\sin^2x=1$ and $\cos^2 x-\sin^2x=\cos 2x$: $=1*\cos 2x$ $=\cos 2x$ Since this equals the right side, the identity is proven.