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

Options

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

Explanation

To solve the given summation problem, we need to evaluate:

\[ S = \sum_{k=1}^{13} \frac{1}{\sin\left(\frac{\pi}{4} + (k-1)\frac{\pi}{6}\right) \sin\left(\frac{\pi}{4} + k\frac{\pi}{6}\right)} \]

### Step 1: Rewrite the terms in the summation
We start by rewriting the sine terms in the summation. We can express the sine terms using the angle addition formula.
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