Question

If a1, a2,....an are in AP with common difference d,then find the sum of the series: sind[coseca1coseca2+coseca2coseca3...cosecan-1cosecan]

Answer 1

If $a_1,a_2,...,a_n$ are in AP, then

$a_2-a_1=d\\ a_3-a_2=d\\ a_4-a_3=d\\ ....................\\ ....................\\ a_n-a_{n-1}=d$.

Now,

$\sin d(\csc a_1\csc a_2+\csc a_2\csc a_3+...+\csc a_{n-1}\csc a_n)=\frac{\sin d}{\sin a_1\sin a_2}+\frac{\sin d}{\sin a_2\sin a_3}+...+\frac{\sin d}{\sin a_{n-1}\sin a_n}\\ \sin d(\csc a_1\csc a_2+\csc a_2\csc a_3+...+\csc a_{n-1}\csc a_n)=\frac{\sin (a_2-a_1)}{\sin a_1\sin a_2}+\frac{\sin (a_3-a_2)}{\sin a_2\sin a_3}+...+\frac{\sin (a_n-a_{n-1})}{\sin a_{n-1}\sin a_n}$

$\frac{\sin (a_n-a_{n-1})}{\sin a_{n-1}\sin a_n}=\frac{\sin a_n\cos a_{n-1}-\cos a_n\sin a_{n-1}}{\sin a_{n-1}\sin a_n}=\cot a_{n-1}-\cot a_{n}$

Thus the sum,

$\sin d(\csc a_1\csc a_2+\csc a_2\csc a_3+...+\csc a_{n-1}\csc a_n)=\cot a_{1}-\cot a_{2}+\cot a_{2}-\cot a_{3}+...+\cot a_{n-1}-\cot a_{n}$

The above series is a telescoping series,

$\sin d(\csc a_1\csc a_2+\csc a_2\csc a_3+...+\csc a_{n-1}\csc a_n)=\cot a_{1}-\cot a_{n}$