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

# If  ${a}_{1},{a}_{2},\dots ,{a}_{n}$ are in A.P., with common difference  $d\ne 0$ then sum of the series $\mathrm{sin}d\left[\mathrm{cosec}{a}_{1}\mathrm{cosec}\right{a}_{2}+\mathrm{cosec}{a}_{2}\mathrm{cosec}{a}_{3}+\dots +\mathrm{cosec}{a}_{n-1}\mathrm{cosec}{a}_{n}]$is

1. A

$\mathrm{sec}{a}_{1}-\mathrm{sec}{a}_{n}$

2. B

$\mathrm{cosec}{a}_{1}-\mathrm{cosec}{a}_{n}$

3. C

$\mathrm{cot}{a}_{1}-\mathrm{cot}{a}_{n}$

4. D

$\mathrm{tan}{a}_{1}-\mathrm{tan}{a}_{n}$

Register to Get Free Mock Test and Study Material

+91

Live ClassesRecorded ClassesTest SeriesSelf Learning

Verify OTP Code (required)

I agree to the terms and conditions and privacy policy.

### Solution:

$\mathrm{sin}d\left[\mathrm{cosec}{a}_{1}\mathrm{cosec}{a}_{2}+\mathrm{cosec}{a}_{2}\mathrm{cosec}{a}_{3}+\cdots +\right\mathrm{cosec}{a}_{n-1}\mathrm{cosec}{a}_{n}]$

$\begin{array}{l}=\frac{\mathrm{sin}\left({a}_{2}-{a}_{1}\right)}{\mathrm{sin}{a}_{1}\mathrm{sin}{a}_{2}}+\frac{\mathrm{sin}\left({a}_{3}-{a}_{2}\right)}{\mathrm{sin}{a}_{2}\mathrm{sin}{a}_{3}}+\cdots +\frac{\mathrm{sin}\left({a}_{n}-{a}_{n-1}\right)}{\mathrm{sin}{a}_{n-1}\mathrm{sin}{a}_{n}}\\ =\left(\mathrm{cot}{a}_{1}-\mathrm{cot}{a}_{2}\right)+\left(\mathrm{cot}{a}_{2}-\mathrm{cot}{a}_{3}\right)\\ +\cdots +\left(\mathrm{cot}{a}_{n-1}-\mathrm{cot}{a}_{n}\right)\\ =\mathrm{cot}{a}_{1}-\mathrm{cot}{a}_{n}\end{array}$

Talk to our academic expert!

+91

Live ClassesRecorded ClassesTest SeriesSelf Learning

Verify OTP Code (required)

I agree to the terms and conditions and privacy policy.