Question

How to prove `cos^2(pi/8)+cos^2 ((3pi)/8)+cos^2((5pi)/8)+cos^2((7pi)/8)=2` ?

Answer 1

Please see below

Explanation:

Let `pi/8=x` then `8x=pi and 4x=pi/2`

`LHS=cos^2(pi/8)+cos^2((3pi)/8)+cos^2((5pi)/8)+cos^2((7pi)/8)`

`=cos^2x+cos^2(3x)+cos^2(5x)+cos^2(7x)`

`=cos^2x+cos^2(3x)+cos^2(4x+x)+cos^2(4x+3x)`

`=cos^2x+cos^2(3x)+cos^2(pi/2+x)+cos^2(pi/2+3x)`

`=cos^2x+cos^2(3x)+sin^2(x)+sin^2(3x)=1+1=2=RHS`