The value of `sum_(r=1)^(11)tan^(2)((r pi)/(24))` is

Options

A. 91
B. 85
C. 253/3
D. none of these

Answer

C

Explanation

`S= sum_(r=1)^(11)tan^(2)((r pi)/(24))`
`=("tan"^(2)(pi)/(24)+"tan"^(2)(11pi)/(24))+("tan"^(2)(2pi)/(24)+"tan"^(2)(10pi)/(24))+("tan"^(2)(3pi)/(24)+"tan"^(2)(9pi)/(24))+("tan"^(2)(4pi)/(24)+"tan"^(2)(8pi)/(24))+("tan"^(2)(5pi)/(24+"tan"^(2)(7pi)/(24))+("tan"^(2)(6pi)/(24))`
`=("tan"^(2)(pi)/(24)+"cot"^(2)(pi)/(24))+{(2-sqrt(3))^(2)+(2+sqrt(3))^(2)+(sqrt(2)-1)^(2)+(sqrt(2)+1)^(2)+((1)/(sqrt(3)))^(2)+(sqrt(3))^(2)}+("tan"^(2)(5pi)/(24)+"cot"^(2)(5pi)/(24))+1`
Now `(tan^(2) theta + cot^(2)theta)=2+4 cot^(2)2theta`
`S=2+4"cot"^(2)(pi)/(12)+(70)/(3)+2+4"cot"^(2)(5pi)/(12)+1`
`=(85)/(3)+4[(2+sqrt(3))^(2)+(2-sqrt(3))^(2)]`
`therefore = (85)/(3)+4(2)(4+3)=(253)/(3)`