Sum the following: tan x+2tan 2x+2^2 tan 2^2 x+...+2^n tan 2^n x


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Question Number 13732 by prakash jain last updated on 22/May/17

$Sum the following :$ $\tan x + 2 \tan 2 x + 2^{2} \tan 2^{2} x + . . . + 2^{n} \tan 2^{n} x$
Commented by ajfour last updated on 23/May/17

$a g e n u i n e s o l u t i o n p l e a s e . .$
Commented by prakash jain last updated on 23/May/17

$I solved using calculus$ $- \int S = \ln \cos x + \ln \cos 2^{2} x + . . . + \ln \cos 2^{n} x + C$ $= \ln \frac{\sin x \cdot \cos x \cdot \cos 2 x \cdot . . . \cdot \cos 2^{n} x}{\sin x} + C$ $= \ln \frac{\sin 2 x \cdot \cos 2 x \cdot \cos 4 x . . . \cos 2^{n} x}{2 \sin x} + C$ $= \ln \frac{\sin 2^{n + 1} x}{2^{n + 1} \sin x} = \ln \sin 2^{n + 1} x - \ln \sin x + C$ $S = - (2^{n + 1} \frac{\cos 2^{n + 1} x}{\sin 2^{n + 1} x} - \frac{\cos x}{\sin x})$ $= \cot x - 2^{n + 1} \cot 2^{n + 1} x$
Commented by ajfour last updated on 23/May/17

$A m a z i n g!$
	Answered by Tinkutara last updated on 23/May/17

	$= \tan x + 2 \tan 2 x + 2^{2} \tan 2^{2} x + . . . + 2^{n} \tan 2^{n} x$ $+ 2^{n + 1} {\cot 2}^{n + 1} x - 2^{n + 1} {\cot 2}^{n + 1} x$ $= \cot x - 2^{n + 1} \cot 2^{n + 1} x$ $[∵ \tan x + 2 \tan 2 x + 2^{2} {\tan 2}^{2} x + . . . +$ $2^{n} {\cot 2}^{n} x = \cot x]$
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