Question

How do you use the half angle identity to find exact value of cos 22.5?

Answer 1

Refer to explanation

Explanation:

We have that

#cos(2a)=cos^2a-sin^2a=cos^2(a)-(1-cos^2a)=> cos^2(a)=1/2*(1+cos2a)=> cos(a)=1/sqrt2*sqrt(1+cos2a)#

Hence set `a=22.5^o` to the last equation we get

`cos(22.5)=sqrt2/2*(sqrt(1+cos45))=(sqrt(2+sqrt2))/2`

Remarks
1. We know that `cos45^o=sqrt2/2`