Answer
$$\cos^2\frac{\pi}{8}-\frac{1}{2}=\frac{\sqrt2}{4}$$
Work Step by Step
$$\cos^2\frac{\pi}{8}-\frac{1}{2}$$
Recall the Double-Angle Identity for cosine:
$$2\cos^2A-1=\cos2A$$
So apparently, this case does not follow the already known Double-Angle Identity for cosine.
That, nevertheless, does not mean we cannot apply the identity. What we need here are some transformations.
$$\cos^2\frac{\pi}{8}-\frac{1}{2}=\Big(2\times\frac{1}{2}\times\cos^2\frac{\pi}{8}-\frac{1}{2}\Big)$$
$$\cos^2\frac{\pi}{8}-\frac{1}{2}=\frac{1}{2}\Big(2\cos^2\frac{\pi}{8}-1\Big)$$
Now $2\cos^2\frac{\pi}{8}-1$ can be applied with the identity $2\cos^2A-1=\cos2A$ for $A=\frac{\pi}{8}$.
$$\cos^2\frac{\pi}{8}-\frac{1}{2}=\frac{1}{2}\Big[\cos\Big(2\times\frac{\pi}{8}\Big)\Big]$$
$$\cos^2\frac{\pi}{8}-\frac{1}{2}=\frac{1}{2}\Big(\cos\frac{\pi}{4}\Big)$$
$$\cos^2\frac{\pi}{8}-\frac{1}{2}=\frac{1}{2}\times\frac{\sqrt2}{2}$$
$$\cos^2\frac{\pi}{8}-\frac{1}{2}=\frac{\sqrt2}{4}$$
