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The value of overset(13)underset(k=1)(su...

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

A

`3-sqrt(3)`

B

`2(3-sqrt(3))`

C

`2(3-sqrt(3))`

D

`2(sqrt(3)-1))`

Text Solution

Verified by Experts

The correct Answer is:
C
`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)`
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