find the sum of the series

Question

If $a_1, a_2, \ldots, a_n$ are in arithmetic progression whose common difference is $d$,then find the sum:

$$\sin(d) \cdot \left(\csc(a_1)\csc (a_2)+\csc(a_2)\csc (a_3)+\ldots+\csc(a_{n-1})\csc(a_n) \right)$$

Answer 1

HINT:

$$\sin d\cdot(\csc a_r\cdot\csc a_{r+1})=\frac{\sin(a_{r+1}- a _r)}{\sin a_r\sin a_{r+1}}=\cot a_r-\cot a_{r+1}$$ where $1\le r\le n-1$

Reference: Telescoping series