Answer
$\frac{1}{4}(2-\sqrt 3)$
Work Step by Step
$ cos^2\frac{5\pi}{12}=\frac{1}{2}(cos\frac{5\pi}{6}+1)=\frac{1}{2}(1-\frac{\sqrt 3}{2})=\frac{1}{4}(2-\sqrt 3)$
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