If a_1, a_2, a_3, dots, a_n are in an arithme | vedclass

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(C) Given that $a_1, a_2, a_3, \dots, a_n$ are in an arithmetic progression,the common difference is $d = a_{k+1} - a_k$ for any $k \in \{1, 2, \dots,

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$\cot a_1 - \cot a_n$

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(C) Given that $a_1, a_2, a_3, \dots, a_n$ are in an arithmetic progression,the common difference is $d = a_{k+1} - a_k$ for any $k \in \{1, 2, \dots, n-1\}$.Consider the expression $S = \sin d [\csc a_1 \csc a_2 + \csc a_2 \csc a_3 + \dots + \csc a_{n-1} \csc a_n]$.Since $\sin d = \sin(a_{k+1} - a_k)$,we can rewrite each term as:$\sin(a_{k+1} - a_k) \csc a_k \csc a_{k+1} = \frac{\sin(a_{k+1} - a_k)}{\sin a_k \sin a_{k+1}}$.Using the identity $\sin(x - y) = \sin x \cos y - \cos x \sin y$,we get:$\frac{\sin a_{k+1} \cos a_k - \cos a_{k+1} \sin a_k}{\sin a_k \sin a_{k+1}} = \frac{\sin a_{k+1} \cos a_k}{\sin a_k \sin a_{k+1}} - \frac{\cos a_{k+1} \sin a_k}{\sin a_k \sin a_{k+1}} = \cot a_k - \cot a_{k+1}$.Summing these terms from $k=1$ to $n-1$ creates a telescoping series:$S = (\cot a_1 - \cot a_2) + (\cot a_2 - \cot a_3) + \dots + (\cot a_{n-1} - \cot a_n)$.All intermediate terms cancel out,leaving $S = \cot a_1 - \cot a_n$.

If $a_1, a_2, a_3, \dots, a_n$ are in an arithmetic progression with common difference $d$,then find the value of $\sin d [\csc a_1 \csc a_2 + \csc a_2 \csc a_3 + \dots + \csc a_{n-1} \csc a_n]$.

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