#### Answer

$\sec^2\frac{x}{2}=\frac{2}{1+\cos x}$

#### Work Step by Step

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.