Evaluate ∑ 1/sin[π/4+(r-1)π/6]sin[π/4+rπ/6] r ∈ to (r=1, 13)

Question

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

(1) \(2 \sqrt{3}+1\)

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

(3) \(3 \sqrt{2}+2\)

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

Answer 1

Correct option is (2) \(2 \sqrt{3}-1\)    

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

   

\(\sin (A-B)=\sin A \cos B-\cos A \sin B\)  

\(=2 \cdot \sum_{r=1}^{13} \cot \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right)-\cot \left(\frac{\pi}{4}+r \cdot \frac{\pi}{6}\right) \)   

\( =2 \cdot \cot \left(\frac{\pi}{4}+0 \cdot \frac{\pi}{6}\right)-\cot \left(\frac{\pi}{4}+\frac{13 \pi}{6}\right) \)  

\(=2 \cdot\left\{\cot \frac{\pi}{4}-\cot \left(\frac{\pi}{4}+\frac{\pi}{6}\right)\right\} \)  

\(=2[1-2+\sqrt{3}] \)   

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