Prove that: `cos^4pi/8+cos^4(3pi)/8+cos^4(5pi)/8+cos^4(7pi)/8=3/2`

Explanation

We have `(7pi)/(8)=pi-(pi)/(8)` and `(5pi)/(8)=pi-(3pi)/(8)`
`rArr cos""(7pi)/(8)=-cos""(pi)/(8)` and `cos""(5pi)/(8)=-cos""(3pi)/(8)`
`rArr cos^(4)""(7pi)/(8)=cos^(4)""(pi)/(8)` and `cos^(4)""(5pi)/(8)=cos^(4)""(3pi)/(8)`
`therefore LHS=2cos^(4)""(pi)/(8)+2cos^(4)""(3pi)/(8)`
`=2[(cos^(2)""(pi)/(8))^(2)+(cos^(2)""(3pi)/(8))^(2)]`
`=2{(1+cos""(pi)/(4))/(2)}^(2)`
`[because (1+cos 2theta)/(2)=cos^(2)theta]`
`=(1)/(2){(1+cos""(pi)/(4))^(2)+(1+cos""(3pi)/(4))^(2)}^(2)`
`=(1)/(2){(1+(1)/sqrt(2))^(2)+(1-(1)/sqrt(2))^(2)}^(2)}`
`=(1)/(2){(1+(1)/(2)+sqrt(2))+(1+(1)/(2)-sqrt(2))}`
`=(3)/(2)=` RHS.