Why do the sum-of-angles and half-angle formulas return different answers for $\cos\frac\pi{12}$? Suppose I want to calculate $\cos\frac{\pi}{12}$. Using the sum-of-angles formula and using the half-angle formula results in different answers.
$$
\cos\frac{\pi}{12} = \cos\frac{\pi/6}{2} = \sqrt{\frac{1}{2}+\frac{\cos\frac{\pi}{6}}{2}} = \sqrt{\frac{1}{2}+\frac{\sqrt{3}}{4}} 
$$$$
\cos\frac{\pi}{12} = \cos\frac{\pi}{3}\cos\frac{\pi}{4}+\sin\frac{\pi}{3}\sin\frac{\pi}{4} = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)+\left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}+\sqrt{6}}{4}
$$
Why does this happen? Shouldn't these be the same?

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Those may not look the same, but they are equal! Notice the first one is wrapped completely in a square root, so rewrite the second one so that it too is wrapped in a single square root:
$$
\frac{\sqrt{2}+\sqrt{6}}{4} = 
\sqrt{\left(\frac{\sqrt{2}+\sqrt{6}}{4}\right)^2} = 
\sqrt{\frac{2 + 2\sqrt{2}\sqrt{6} + 6}{16}} = 
\sqrt{\frac{8 + 4\sqrt{3}}{16}} = 
\sqrt{\frac{1}{2} + \frac{\sqrt{3}}{4}} 
$$
