If  a1,a2,…,an are in A.P., with common difference  d≠0 then sum of the series sin⁡dcosec⁡a1coseca2+cosec⁡a2cosec⁡a3+…+cosec⁡an−1cosec⁡anis

sin⁡dcosec⁡a1cosec⁡a2+cosec⁡a2cosec⁡a3+⋯+cosec⁡an−1cosec⁡an=sin⁡a2−a1sin⁡a1sin⁡a2+sin⁡a3−a2sin⁡a2sin⁡a3+⋯+sin⁡an−an−1sin⁡an−1sin⁡an=cot⁡a1−cot⁡a2+cot⁡a2−cot⁡a3+⋯+cot⁡an−1−cot⁡an=cot⁡a1−cot⁡an