if $f(x)= \sum_{k=0}^{\infty} (cos^2x)^k$, then $f(\frac{\pi}{4})$ is
2. $\cos^2(\frac\pi4) = \frac12$.
3. So this is simply the geometric sum $\sum_{k=0}^\infty \left(\frac{1}{2}\right)^2$