Question

`(1+cos.(pi)/(8))(1+cos.(3pi)/(8))(1+cos.(5pi)/(8))(1+cos.(7pi)/(8))` is equal to
A. `(1)/(2)`
B. `cos.(pi)/(8)`
C. `(1)/(8)`
D. `(1+sqrt(2))/(2sqrt(2))`

Answer 1

Correct Answer - C
Given expression = `(1+cos.(pi)/(8))(1+cos.(3pi)/(8))(1+cos.(5pi)/(8))(1+cos.(7pi)/(8))`
`(1+cos.(pi)/(8))(1+cos.(3pi)/(8))(1-cos.(3pi)/(8))(1-cos.(pi)/(8))`
`=(1-cos^(2).(pi)/(8))(1-cos^(2).(3pi)/(8))`
`=(1)/(4)(2-1-cos.(pi)/(4))(2-1-cos3.(pi)/(4))`
`=(1)/(4)(1-cos.(pi)/(4))(1-cos3.(pi)/(4))`
`=(1)/(4)(1-(1)/(sqrt(2)))(1+(1)/(sqrt(2)))=(1)/(4)(1-(1)/(2))=(1)/(8)`