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If the expression cos^2(pi/11)+cos^2((2p...

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`

Text Solution

Verified by Experts

The correct Answer is:
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)`
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