1. Show that

   $$\cos 2\theta =1-2{\sin }^2\theta$$

   and hence prove that

   $$\sin \left(\frac{\pi }{8}\right)=\frac{1}{2}\sqrt{2-\sqrt{2}}$$

2. Show that

   $$\cos 2\theta =2{\cos }^2\theta -1$$

   and hence find the exact value of

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

For (a), we have

and also

Substitute $\theta =\frac{\pi }{8}$ into the identity

and remember that $\cos \left(\frac{\pi }{4}\right)=\frac{\sqrt{2}}{2}.$