## Precalculus (6th Edition)

$\sec^2\frac{x}{2}=\frac{2}{1+\cos x}$
Start with the left side: $\sec^2\frac{x}{2}$ Rewrite secant as the inverse function of cosine: $=\frac{1}{\cos^2\frac{x}{2}}$ Expand using the half-angle identity for cosine: $=\frac{1}{\left(\pm\sqrt\frac{1+\cos x}{2}\right)^2}$ Simplify: $=\frac{1}{\frac{1+\cos x}{2}}$ $=\frac{1}{\frac{1+\cos x}{2}}*\frac{2}{2}$ $=\frac{2}{1+\cos x}$ Since this equals the right side, the identity has been proven.