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The value of sum(k=1)^(13) (1)/(sin((pi)...

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

A

` 3-sqrt3`

B

`2(3-sqrt3)`

C

`2(sqrt3-1)`

D

`2(2+sqrt3)`

Text Solution

Verified by Experts

The correct Answer is:
C
`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)`
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