<s>[INST] Exact value of $\cos^2(\frac{\pi}{8})+\sin^2(\frac{15\pi}{8})?$ I tried separating it into $\cos^2(\frac{\pi}{8})+\sin^2(\frac{\pi}{8}+\frac{14\pi}{8})?$ and using the angle sum identity but it didn't help.
 [/INST] Notice that $\frac{15\pi}{8} = 2\pi - \frac{\pi}{8}$. So $\sin \frac{15\pi}{8} = -\sin \frac{\pi}{8}$. Your expression then reduces/evaluates down to: 
$$\cos^2 \frac{\pi}{8} + \sin^2 \frac{\pi}{8} = 1$$
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