Question

Question is 15 points

Answer 1

$\dfrac{\pi}{12}=\dfrac{1}{12}\pi=\dfrac{4}{12}\pi-\dfrac{3}{12}\pi=\dfrac{1}{3}\pi-\dfrac{1}{4}\pi\\\\\dfrac{\pi}{12}=\dfrac{\pi}{3}-\dfrac{\pi}{4}$

$\cos\dfrac{\pi}{12}=\cos\left(\dfrac{\pi}{3}-\dfrac{\pi}{4}\right)=(*)$

$Use:\ \cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)$

$(*)=\cos\dfrac{\pi}{3}\cos\dfrac{\pi}{4}+\sin\dfrac{\pi}{3}\sin\dfrac{\pi}{4}\\\\=\dfrac{1}{2}\cdot\dfrac{\sqrt2}{2}+\dfrac{\sqrt3}{2}\cdot\dfrac{\sqrt2}{2}\\\\=\dfrac{\sqrt2}{4}+\dfrac{\sqrt6}{4}=\dfrac{\sqrt2+\sqrt6}{4}$