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Question: If <img src="https://cdn.pureessence.tech/canvas_436.png?top_left_x=1444&top_left_y=1769&width=300&h...

If are in A.P. with common difference , , then the sum of the following series is

sin⁡d(cosec⁡a1⋅cosec⁡a2+cosec⁡a2⋅cosec⁡a3+………\sin d \left( \operatorname { cosec } a _ { 1 } \cdot \operatorname { cosec } a _ { 2 } + \operatorname { cosec } a _ { 2 } \cdot \operatorname { cosec } a _ { 3 } + \ldots \ldots \ldots \right.

+cosec⁡an−1cosec⁡an)\left. + \operatorname { cosec } a _ { n - 1 } \operatorname { cosec } a _ { n } \right)

A

sec⁡a1−sec⁡an\sec a _ { 1 } - \sec a _ { n }

B

cot⁡a1−cot⁡an\cot a _ { 1 } - \cot a _ { n }

C

tan⁡a1−tan⁡an\tan a _ { 1 } - \tan a _ { n }

D

cosec⁡a1−cosec⁡an\operatorname { cosec } a _ { 1 } - \operatorname { cosec } a _ { n }

Answer

cot⁡a1−cot⁡an\cot a _ { 1 } - \cot a _ { n }

Explanation

Solution

As given

d=a2−a1=a3−a2=….=an−an−1d = a _ { 2 } - a _ { 1 } = a _ { 3 } - a _ { 2 } = \ldots . = a _ { n } - a _ { n - 1 }

∴\therefore sin⁡d{cosec⁡a1cosec⁡a2+…..+cosec⁡an−1cosec⁡an}\sin d \left\{ \operatorname { cosec } a _ { 1 } \operatorname { cosec } a _ { 2 } + \ldots . . + \operatorname { cosec } a _ { n - 1 } \operatorname { cosec } a _ { n } \right\}

=(cot⁡a1−cot⁡a2)+(cot⁡a2−cot⁡a3)+…= \left( \cot a _ { 1 } - \cot a _ { 2 } \right) + \left( \cot a _ { 2 } - \cot a _ { 3 } \right) + \ldots +(cot⁡an−1−cot⁡an)+ \left( \cot a _ { n - 1 } - \cot a _ { n } \right)

=cot⁡a1−cot⁡an= \cot a _ { 1 } - \cot a _ { n }.