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. (a) `3-sqrt(3)`
B. (b)`2(3-sqrt(3))`
C. (c) `2(sqrt(3)-1)`
D. (d) `2(2+sqrt3)`

Answer

(c)

Explanation

Here, `sum_(k=1)^(13)1/(sin{pi/4+((k-1)pi)/6}sin (pi/4+(kpi)/6))`
Converting into differences, by multiplying and
dividing by `sin [(pi/4+(kpi)/6)-{pi/4+((k-1)pi)/6}],i.e.sin (pi/6).`
`therefore sum_(k=1)^(13)(sin[(pi/4=kpi/6)-{pi/4+(k-1)pi/6}])/(sin frac {pi}{6} {sin{pi/4+(k-1)pi/6}sin (pi/4+kpi/6)})`
`=2sum_(k=1)^13 ([sin(pi/4+(kpi)/6)cos{pi/4+(k-1)pi/6} -sin{pi/4+(k-1)pi/6}cos(pi/4+(kpi)/6)])/(sin{pi/4+(k-1)pi/6}-cot (pi/4+kpi/6))`
`=2sum_(k=1)^13 [cot{pi/4+(k-1)pi/6}-cot(pi/4+k pi/6)]`
`=2 [{cot(pi/4)-cot(pi/4+ pi/6)}`
`+ {cot(pi/4+pi/6)-cot(pi/4+ (2pi)/6)}`
`+...+ {cot(pi/4+12 pi/6)-cot(pi/4+ 13pi/6)}]`
`= 2 {cot frac{pi}{4}-cot(pi/4+ 13pi/6)}`
`= 2 [1-cot ((29pi)/12)]=2 [1-cot(2pi+(5pi)/12)]`
`=2[1-cot frac{5pi}{12}] [therefore cot frac {5pi}{12}=(2-sqrt3)]`
`=2(1-2+sqrt(3))`
`=2(sqrt(3)-1)`