## Precalculus (6th Edition)

$\cos 2\theta=\frac{2-\sec^2 \theta}{\sec^2 \theta}$
Start with the right side: $\frac{2-\sec^2 \theta}{\sec^2 \theta}$ Break it into two fractions and simplify: $=\frac{2}{\sec^2 \theta}-\frac{\sec^2 \theta}{\sec^2 \theta}$ $=\frac{2}{\frac{1}{\cos^2\theta}}-1$ $=2\cos^2\theta-1$ $=\cos 2\theta$ Since this equals the left side, the identity has been proven.