Prove that: \[ \cos \frac{\pi}{12} – \sin \frac{\pi}{12} = \frac{1}{\sqrt{2}} \]
Solution
Using the identity:
\[
\cos \theta – \sin \theta
=
\sqrt{2}\cos\left(\theta+ \frac{\pi}{4}\right)
\]
Taking
\[
\theta = \frac{\pi}{12}
\]
Then,
\[
\cos \frac{\pi}{12} – \sin \frac{\pi}{12}
=
\sqrt{2}\cos\left(\frac{\pi}{12}+\frac{\pi}{4}\right)
\]
\[
=
\sqrt{2}\cos\left(\frac{\pi}{12}+\frac{3\pi}{12}\right)
\]
\[
=
\sqrt{2}\cos\frac{4\pi}{12}
\]
\[
=
\sqrt{2}\cos\frac{\pi}{3}
\]
\[
=
\sqrt{2}\times\frac{1}{2}
\]
\[
=
\frac{1}{\sqrt{2}}
\]
Hence,
\[
\boxed{
\cos \frac{\pi}{12} – \sin \frac{\pi}{12}
=
\frac{1}{\sqrt{2}}
}
\]