Answer
$$\sec^2\frac{x}{2}=\frac{2}{1+\cos x}$$
The equation is verified to be an identity below.
Work Step by Step
$$\sec^2\frac{x}{2}=\frac{2}{1+\cos x}$$
We start with the left side $$A=\sec^2\frac{x}{2}$$ $$A=\frac{1}{\cos^2\frac{x}{2}}$$ We know that $\cos2x=2\cos^2x-1$, which means $\cos^2x=\frac{\cos 2x+1}{2}$. Therefore, $\cos^2\frac{x}{2}$ can also be written in the same way: $\cos^2\frac{x}{2}=\frac{\cos x+1}{2}$
Therefore, $$A=\frac{1}{\frac{\cos x+1}{2}}$$ $$A=\frac{2}{\cos x+1}$$
The equation has been proved to be an identity.
