The values of `sum_(k=1)^(13) (1)/(sin((pi)/(4)+(k-1)(pi)/(6))sin((pi)/(4)+(kn)/(6)))` is equal

Options

A. `3-sqrt3`
B. `2(3-sqrt3)`
C. `2(3-sqrt3)`
D. `2(2+sqrt3)`

Answer

C

Explanation

`underset(k=1)overset(13)(sum)(1)/(sin((pi)/(4)+(k-1)(pi)/(6))sin((pi)/(4)+(kpi)/(6)))`
`=2 underset(k=1)overset(13)(sum)(sin {((pi)/(4)++(kpi)/(6))-((pi)/(4)+(k-1)(pi)/(6))})/(sin((pi)/(4)+(k-1)(pi)/(6)(pi)/(6))-sin((pi)/(4)+(kpi)/(6)))`
`=2 underset(k=1)overset(13)(sum){cot((pi)/(4)+(k-1)(pi)/(6))-cot((pi)/(4)+(kpi)/(6))}`
`=2{cot""(pi)/(4)cot((pi)/(4)+(13pi)/(6))}=2(1-cot""(29pi)/(12))=2(1-cot""(5pi)/(12))`
`=2{1-(2-sqrt3)}=2(sqrt3-1)`