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

Options

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

Answer

C

Explanation

`2sum_(k=1)^(13)(sin(pi/6))/(sin(pi/4+((k-1)pi)/6)sin(pi/4+(kpi)/6))`
`=2sum(sin{(pi/4+(kpi)/6)-(pi/4+((k-1)pi)/6)})/(sin(pi/4+((k-1)pi)/6)cdotsin(pi/4+(kpi)/6))`
`=2sum_(k=1)^(13){cot(pi/4+((k-1)pi)/6)-cot(pi/4+(kpi)/6)}`
`=2[cot(pi/4)-cot(pi/4+(13pi)/6)]`
`=2[1-(2-sqrt3)]`
`=2(sqrt3-1)`