Question

How do you evaluate $2{\cos }^2\left(\frac{\pi }{12}\right)-1$?

Answer 1

We know that $\cos \left(2x\right)=2{\cos }^2\left(x\right)-1$

Hence for $x=\frac{\pi }{12}$ we have that

$2{\cos }^2\left(\frac{\pi }{12}\right)-1=\cos (2\cdot \frac{\pi }{12})=\cos \left(\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}$.