Question

Prove that $\cos \frac{\pi }{12}-\sin \frac{\pi }{12}=\frac{1}{\sqrt{2}}$

Answer 1

$cos\left(\frac{\pi }{12}\right)-sin\left(\frac{\pi }{12}\right)$

$=\sqrt{2}\left[\frac{1}{\sqrt{2}}cos\right(\frac{\pi }{12})-\frac{1}{\sqrt{2}}sin(\frac{\pi }{12}\left)\right]$

$=\sqrt{2}\left[cos\right(\frac{\pi }{4}\left)cos\right(\frac{\pi }{12})-sin(\frac{\pi }{4}\left)sin\right(\frac{n}{12}\left)\right]$

$=\sqrt{2}cos\left(\frac{\pi }{4}\right)+\frac{\pi }{12}$

$=\sqrt{2}cos\left(\frac{4\pi }{12}\right)=\sqrt{2}cos\left(\frac{\pi }{3}\right)=\sqrt{2}\frac{1}{2}=\frac{1}{\sqrt{2}}$

Hence $cos\left(\frac{\pi }{12}\right)-sin\left(\frac{\pi }{12}\right)=\frac{1}{\sqrt{2}}$