The sum of all the solutions of the equation 1 + 2 cosec x = −((sec^2 (x/2))/2) in the interval [0, 4π] is nπ, where n is equal to


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Question Number 22348 by Tinkutara last updated on 16/Oct/17

$The sum of all the solutions of the$ $equation 1 + 2 cosec x = - \frac{\sec^{2} \frac{x}{2}}{2} in$ $the interval [0, 4 π] is n π, where n is$ $equal to$
	Answered by ajfour last updated on 16/Oct/17

	$n = 5, a s t h e r e a r e t w o$ $p o s s i b l e s o l u t i o n s i n t h e g i v e n$ $i n t e r v a l . x = \frac{3 π}{2}, \frac{7 π}{2} .$ $L e t \tan \frac{x}{2} = t$ $2 cosec x = \frac{2}{\sin x} = \frac{1 + t^{2}}{t}$ $\sec^{2} \frac{x}{2} = 1 + t^{2}; T h e n$ $1 + \frac{1 + t^{2}}{t} = - \frac{(1 + t^{2})}{2}$ $2 t + 2 + 2 t^{2} + t (1 + t^{2}) = 0$ $o r t^{3} + 2 t^{2} + 3 t + 2 = 0$ $\Rightarrow (t + 1) (t^{2} + t + 2) = 0$ $o n l y o n e r e a l r o o t t = - 1$ $\Rightarrow \tan (\frac{x}{2}) = - 1$ $A s 0 ⩽ \frac{x}{2} ⩽ 2 π$ $\Rightarrow \frac{x}{2} = \frac{3 π}{4}, \frac{7 π}{4}$ $\Rightarrow x = \frac{3 π}{2}, \frac{7 π}{2}$ $T h e i r s u m = 5 π = n π$ $S o n = 5 .$
	Commented by Tinkutara last updated on 16/Oct/17

	$Thank you very much Sir!$
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