If {x_1}, {x_2}, {x_3}, dots, {x_n} are in A. | vedclass

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(A) Given that ${x_1}, {x_2}, \dots, {x_n}$ are in $A.P.$ with common difference $\alpha$,we have ${x_{k+1}} - {x_k} = \alpha$ for all $k = 1, 2, \dot

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$\frac{\sin (n-1)\alpha}{\cos {x_1} \cos {x_n}}$

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(A) Given that ${x_1}, {x_2}, \dots, {x_n}$ are in $A.P.$ with common difference $\alpha$,we have ${x_{k+1}} - {x_k} = \alpha$ for all $k = 1, 2, \dots, n-1$.We can write the expression as:$S = \sum_{k=1}^{n-1} \sin \alpha \sec {x_k} \sec {x_{k+1}}$Since $\sin \alpha = \sin ({x_{k+1}} - {x_k})$,we have:$S = \sum_{k=1}^{n-1} \frac{\sin ({x_{k+1}} - {x_k})}{\cos {x_k} \cos {x_{k+1}}}$Using the formula $	an A - 	an B = \frac{\sin (A-B)}{\cos A \cos B}$,we get:$S = \sum_{k=1}^{n-1} (	an {x_{k+1}} - 	an {x_k})$This is a telescoping sum:$S = (	an {x_2} - 	an {x_1}) + (	an {x_3} - 	an {x_2}) + \dots + (	an {x_n} - 	an {x_{n-1}})$$S = 	an {x_n} - 	an {x_1} = \frac{\sin ({x_n} - {x_1})}{\cos {x_n} \cos {x_1}}$Since ${x_n} = {x_1} + (n-1)\alpha$,we have ${x_n} - {x_1} = (n-1)\alpha$.Thus,$S = \frac{\sin (n-1)\alpha}{\cos {x_1} \cos {x_n}}$.

If ${x_1}, {x_2}, {x_3}, \dots, {x_n}$ are in $A.P.$ whose common difference is $\alpha$,then the value of $\sin \alpha (\sec {x_1} \sec {x_2} + \sec {x_2} \sec {x_3} + \dots + \sec {x_{n-1}} \sec {x_n}) = $

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