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

Options

A. ` 3-sqrt3`
B. `2(3-sqrt3)`
C. `2(sqrt3-1)`
D. `2(2+sqrt3)`

Answer

C

Explanation

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