Question

If the expression `cos^2(pi/11)+cos^2((2pi)/11)+cos^2((3pi)/11)+cos^2((4pi)/11)+cos^2((5pi)/11)` has the value equal to p/q in it lowest from ; then find `(p+q)`
A. `7//4`
B. `9//4`
C. `11/4`
D. `13//4`

Answer 1

Correct Answer - B
`"cos"^(2)(pi)/(11)+"cos"^(2)(2pi)/(11)+"cos"^(2)(4pi)/(11)+"cos"^(2)(5pi)/(11)`
`=(1)/(2)[(1+cos.(2pi)/(11))+(1+cos.(4pi)/(11))+…+(1+cos.(10pi)/(11))]`
`=(1)/(2)[5+(cos.(2pi)/(11)+cos.(4pi)/(11)+cos.(6pi)/(11)+…+cos.(10pi)/(11))]`
`=(1)/(2)[5+(sin.(5pi)/(11))/(sin.(pi)/(11))cos(((2pi)/(11)+(10pi)/(11))/(2))]`
`=(1)/(2)[5+(sin.(5pi)/(11))/(sin.(pi)/(11))cos.(6pi)/(11)]`
`=(1)/(2)[5-(2sin.(5pi)/(11)cos.(5pi)/(11))/(2sin.(pi)/(11))]`
`=(1)/(2)[5-(sin.(10pi)/(11))/(2sin.(pi)/(11))]`
`=(1)/(2)[5-(1)/(2)]`
`=(9)/(4)`