The value of cos

$\cos \frac{\pi}{12}-\sin \frac{\pi}{12}$

multiply and divide above equation by $\sqrt{2}$

i. e. $\sqrt{2}\left[\frac{1}{\sqrt{2}} \cos \frac{\pi}{12}-\frac{1}{\sqrt{2}} \sin \frac{\pi}{12}\right]$

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

$=\sqrt{2}\left[\cos \left(\frac{\pi}{4}+\frac{\pi}{12}\right)\right]$    [using identify $\cos (a+b)=\cos a \cos b-\sin a \sin b$ ]

$=\sqrt{2}\left[\cos \left(\frac{3 \pi+\pi}{12}\right)\right]$

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

$=\sqrt{2} \times \frac{1}{2}=\frac{1}{\sqrt{2}}$

Hence, value of $\cos \frac{\pi}{12}-\sin \frac{\pi}{12}$ is $\frac{1}{\sqrt{2}}$.