The value of k = 1 13 1 sin ( 4 + ( k - 1 ) 6 ) sin ( 4 + k 6 ) is equal to [2016]

Question

The value of $\sum_{k = 1}^{13} \frac{1}{\sin (\frac{π}{4} + \frac{(k - 1) π}{6}) \sin (\frac{π}{4} + \frac{k π}{6})}$ is equal to [2016]

Smart Achievers · Accepted Answer

(3)

$\sum_{k = 1}^{13} \frac{1}{\sin (\frac{π}{4} + (k - 1) \frac{π}{6}) \sin (\frac{π}{4} + \frac{k π}{6})}$

$= \sum_{k = 1}^{13} \frac{1}{\sin \frac{π}{6}} [\frac{\sin {\frac{π}{4} + \frac{k π}{6} - (\frac{π}{4} + (k - 1) \frac{π}{6})}}{\sin (\frac{π}{4} + (k - 1) \frac{π}{6}) \sin (\frac{π}{4} + \frac{k π}{6})}]$

$= \sum_{k = 1}^{13} 2 [\cot (\frac{π}{4} + (k - 1) \frac{π}{6}) - \cot (\frac{π}{4} + \frac{k π}{6})]$

$= 2 [{\cot \frac{π}{4} - \cot (\frac{π}{4} + \frac{π}{6})} + {\cot (\frac{π}{4} + \frac{π}{6}) - \cot (\frac{π}{4} + \frac{2 π}{6})} + \dots + {\cot (\frac{π}{4} + \frac{12 π}{6}) - \cot (\frac{π}{4} + \frac{13 π}{6})}]$

$= 2 [\cot \frac{π}{4} - \cot (\frac{π}{4} + \frac{13 π}{6})]$ $= 2 [1 - \cot \frac{5 π}{12}]$

$= 2 [1 - \frac{\sqrt{3} - 1}{\sqrt{3} + 1}]$ $= 2 [1 - (2 - \sqrt{3})]$ $= 2 (\sqrt{3} - 1)$

Solution

Q.
The value of $\sum_{k = 1}^{13} \frac{1}{\sin (\frac{π}{4} + \frac{(k - 1) π}{6}) \sin (\frac{π}{4} + \frac{k π}{6})}$ is equal to [2016]

1 $3 - \sqrt{3}$

2 $2 (3 - \sqrt{3})$

3 $2 (\sqrt{3} - 1)$

4 $2 (2 - \sqrt{3})$

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Solution

Q. The value of ∑k=1131sin(π4+(k-1)π6)sin(π4+kπ6) is equal to [2016]

1 3-3

2 2(3-3)

3 2(3-1)