If a1,a2,…,an  are in AP  with common  difference  d, then the  sum of the  series sind cosec ⁡a1cosec ⁡a2+cosec ⁡a2cosec ⁡a3+…+cosec⁡an−1cosec⁡an is

# If ${\mathrm{a}}_{1},{\mathrm{a}}_{2},\dots ,{\mathrm{a}}_{\mathrm{n}}$  are in AP  with common  difference  d, then the  sum of the  series sind $\mathrm{cosec}{\mathrm{a}}_{\mathrm{n}-1}\mathrm{cosec}{\mathrm{a}}_{\mathrm{n}}\right)$ is

1. A

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

2. B

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

3. C

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

4. D

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

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### Solution:

Since,${\mathrm{a}}_{1},{\mathrm{a}}_{2},{\mathrm{a}}_{3},\dots ,{\mathrm{a}}_{\mathrm{n}}$ are in AP

$\begin{array}{l}\therefore \mathrm{sin}\mathrm{d}\left(\mathrm{cosec}{\mathrm{a}}_{1}\mathrm{cosec}{\mathrm{a}}_{2}+\dots +\mathrm{cosec}{\mathrm{a}}_{\mathrm{n}-1}\mathrm{cosec}{\mathrm{a}}_{\mathrm{n}}\right)\\ =\frac{\mathrm{sin}\left({\mathrm{a}}_{2}-{\mathrm{a}}_{1}\right)}{\mathrm{sin}{\mathrm{a}}_{1}\mathrm{sin}{\mathrm{a}}_{2}}+\dots +\frac{\mathrm{sin}\left({\mathrm{a}}_{\mathrm{n}}-{\mathrm{a}}_{\mathrm{n}-1}\right)}{\mathrm{sin}{\mathrm{a}}_{\mathrm{n}-1}\mathrm{sin}{\mathrm{a}}_{\mathrm{n}}}\\ =\frac{\left(\mathrm{sin}{\mathrm{a}}_{2}\mathrm{cos}{\mathrm{a}}_{1}-\mathrm{cos}{\mathrm{a}}_{2}\mathrm{sin}{\mathrm{a}}_{1}\right)}{\mathrm{sin}{\mathrm{a}}_{1}\mathrm{sin}{\mathrm{a}}_{2}}+\dots +\frac{\left(\mathrm{sin}{\mathrm{a}}_{\mathrm{n}}\mathrm{cos}{\mathrm{a}}_{\mathrm{n}-1}+\mathrm{cos}{\mathrm{a}}_{\mathrm{n}}\mathrm{sin}{\mathrm{a}}_{\mathrm{n}-1}\right)}{\mathrm{sin}{\mathrm{a}}_{\mathrm{n}-1}\mathrm{sin}{\mathrm{a}}_{\mathrm{n}}}\end{array}$

$+\dots +\left(\mathrm{cot}{\mathrm{a}}_{\mathrm{n}-1}-\mathrm{cot}{\mathrm{a}}_{\mathrm{n}}\right)$

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

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