Let a 1- a 2- a 3- - be an A-P- with common difference - - 6 and cosec a 1cosec a 2-cosec a 2cosec a 3-cosec a n -1cosec a n - k a 1- an - Then k

Cosec a_1cosec a_2+cosec a_2cosec a_3+...cosec a_n 1cosec a_n =1sin a_1sin a_2+1sin a_2sin a_3+...+1sin a_n 1sin a_n =1sinπ/6[sin(a_2 a_1)sin a_1sin a_2+sin(a_3 ...

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Byju's Answer

Standard XII

Mathematics

Arithmetic Mean

Let a 1, a 2,...

Question

Let $a_{1}, a_{2}, a_{3}, . . .$ be an A.P. with common difference $π / 6$ and $cosec a_{1} cosec a_{2} + cosec a_{2} cosec a_{3} + . . . cosec a_{n - 1} cosec a_{n} = k (cot a_{1} - cot a_{n})$ . Then $k$

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Solution

$cosec a_{1} cosec a_{2} + cosec a_{2} cosec a_{3} + . . . cosec a_{n - 1} cosec a_{n}$
$= \frac{1}{sin a_{1} sin a_{2}} + \frac{1}{sin a_{2} sin a_{3}} + . . . + \frac{1}{sin a_{n - 1} sin a_{n}}$
$= \frac{1}{sin π / 6} [\frac{sin (a_{2} - a_{1})}{sin a_{1} sin a_{2}} + \frac{sin (a_{3} - a_{2})}{sin a_{2} sin a_{3}} + . . . + \frac{sin (a_{n} - a_{n - 1})}{sin a_{n - 1} sin a_{n}}]$
$= 2 [(cot a_{1} - cot a_{2}) + (cot a_{2} - cot a_{3}) + . . . + (cot a_{n - 1} - cot a_{n})]$
$= 2 [(cot a_{1} - cot a_{2})]$

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