`(1+cos.(pi)/(8))(1+cos.(3pi)/(8))(1+cos.(5pi)/(8))(1+cos.(7pi)/(8))` is equal to

Options

A. `(1)/(2)`
B. `(1)/(2)+(1)/(2sqrt(2))`
C. `(1)/(2)-(1)/(2sqrt(2))`
D. `(1)/(8)`

Answer

D

Explanation

`[1+"cos"(pi)/(8)][1+"cos"(3pi)/(8)][1+"cos"(5pi)/(8)][1+(cos7pi)/(8)]`
We have ,
`"cos"(7pi)/(8)=cos[pi-(pi)/(8)]=-"cos"(pi)/(8)`
and `"cos"(5pi)/(8)=cos[pi-(3pi)/(8)]=-"cos"(3pi)/(8)`
`therefore=[1+"cos"(pi)/(8)][1+"cos"(3pi)/(8)][1-"cos"(pi)/(8)][1-"cos"(3pi)/(8)]`
`=[1-"cos"^(2)(pi)/(8)][1-"cos"^(2)(3pi)/(8)]="sin"^(2)(pi)/(8)*"sin"^(2)(3pi)/(8)`
`=(1)/(4)[2"sin"^(2)(pi)/(8)*2"sin"^(2)(3pi)/(8)]`
`=(1)/(4)[(1-"cos"(pi)/(4))(1-"cos"(3pi)/(4))]`
`(because1-costheta=2sin^(2))`
`=(1)/(4)[(1-(1)/(sqrt(2)))(1+(1)/(sqrt(2)))]=(1)/(8)`