Find the value of: (1/(sin 2°))+(1/(sin 4°))+(1/(sin 8°))+(1/(sin 16°))+... +(1/(sin (2^(1029) )^° ))


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Question Number 72447 by Maclaurin Stickker last updated on 28/Oct/19

$F i n d t h e v a l u e o f :$ $\frac{1}{s i n 2 °} + \frac{1}{s i n 4 °} + \frac{1}{s i n 8 °} + \frac{1}{s i n 16 °} + . . .$ $+ \frac{1}{s i n {(2^{1029})}^{°}}$
	Answered by Tanmay chaudhury last updated on 28/Oct/19

	$l e t$ $a = 2^{o} a n d n = 1029$ $S = c o s e c a + c o s e c 2 a + c o s e c 2^{2} a + . . . + c o s e c 2^{n - 1} a$ $n o w c o t a - c o t \frac{a}{2}$ $= \frac{c o s a}{s i n a} - \frac{c o s \frac{a}{2}}{s i n \frac{a}{2}}$ $= \frac{s i n \frac{a}{2} c o s a - c o s \frac{a}{2} s i n a}{s i n a . s i n \frac{a}{2}} = \frac{s i n (\frac{a}{2} - a)}{s i n a . s i n \frac{a}{2}} = - c o s e c a$ $c o s e c a = c o t \frac{a}{2} - c o t a$ $c o s e c 2 a = c o t a - c o t 2 a$ $c o s e c 2^{2} a = c o t 2 a - c o t 2^{2} a$ $. . . .$ $. . . .$ $c o s e c 2^{n - 1} a = c o t 2^{n - 2} a - c o t 2^{n - 1} a$ $a d d t h e m$ $S = c o t \frac{a}{2} - c o t 2^{n - 1} a$ $S = c o t 1^{o} - c o t {(2^{1028})}^{o}$
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