Question

What is `Cos^2(pi/7)+cos^2(2pi/7)+cos^2(4pi/7)` is equal to?

Answer 1

`5/4`

Explanation:

LHS`=cos^2(pi/7)+cos^2(2pi/7)+cos^2(4pi/7)`
using formula `cos^2theta=1/2(1+cos2theta)`

`=1/2(1+cos(2pi/7))+1/2(1+cos(4pi/7))+ 1/2(1+cos(8pi/7))`
`=3/2+1/2(cos(2pi/7)+cos(4pi/7)+cos(8pi/7))`

`=3/2+1/(4sin(pi/7)) (2sin(pi/7)cos(2pi/7)+2sin(pi/7)cos(4pi/7)+2sin(pi/7)cos(8pi/7))`

`=3/2+1/(4sin(pi/7)) (cancel(sin(3pi/7))-sin(pi/7)+sin(5pi/7)-cancel(sin(3pi/7))+sin(9pi/7)-sin(7pi/7))`

`=3/2+1/(4sin(pi/7)) (-sin(pi/7)+sin(pi-2pi/7)+sin(pi+2pi/7)-0)`

`=3/2+1/(4sin(pi/7)) (-sin(pi/7)+cancel(sin(2pi/7))-cancel(sin(2pi/7)))`

`=3/2+1/(4cancel(sin(pi/7))) (-cancel(sin(pi/7)))`

`=3/2-1/4=(6-1)/4=5/4`