If $a_{1}, a_{2}, a_{3}, \dots \dots a_{n}$ are in A.P., with common difference d, then the sum of the series $s i n d [c o s e c a_{1} c o s e c a_{2} + c o s e c a_{1} c o s e c a_{3} + \dots \dots + c o s e c a_{n} - 1 c o s e c a_{n}]$ is

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Solution

We have :

$s i n d [c o s e c a_{1} c o s e c a_{2} + c o s e c a_{2} c o s e c a_{3} + \dots \dots + c o s e c a_{n} - 1 c o s e c a_{n}]$

$= \frac{s i n d}{s i n a_{1} s i n a_{2}} + \frac{s i n d}{s i n a_{2} s i n a_{3}} + \dots \dots + \frac{s i n d}{s i n a_{n - 1} s i n a_{n}}$

$= \frac{s i n (a_{2} - a_{1})}{s i n a_{1} s i n a_{2}} + \frac{s i n (a_{2} - a_{1})}{s i n a_{1} s i n a_{2}} + \dots \dots + \frac{s i n (a_{n} - a_{n - 1})}{s i n a_{n - 1} s i n a_{n}}$

$= \frac{s i n a_{2} c o s a_{1} - c o s a_{2} s i n a_{1}}{s i n a_{1} s i n a_{2}} + \frac{s i n a_{3} c o s a_{2} - c o s a_{3} s i n a_{2}}{s i n a_{1} s i n a_{2}} + \frac{s i n a_{2} c o s a_{1} - c o s a_{2} s i n a_{1}}{s i n a_{1} s i n a_{2}}$

$= (c o t a_{1} - c o t a_{2}) + (c o t a_{2} - c o t a_{3}) + \dots \dots + [(c o t a_{n - 1} - c o t a_{n})]$

$= c o t a_{1} - c o t a_{n}$

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