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