The number of values of x in the interval lef | vedclass

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Question

$x \in\left(\frac{\pi}{4}, \frac{7 \pi}{4}\right)$

$14 \operatorname{cosec}^{2} x-2 \sin ^{2} x=21-4 \cos ^{2} x$

$=21-4\left(1-\sin ^{2} x\righ

Vedclass · Accepted Answer

$4$

Vedclass · Answer

$x \in\left(\frac{\pi}{4}, \frac{7 \pi}{4}ight)$

$14 \operatorname{cosec}^{2} x-2 \sin ^{2} x=21-4 \cos ^{2} x$

$=21-4\left(1-\sin ^{2} xight)$

$=17+4 \sin ^{2} x$

$14 \operatorname{cosec} x-6 \sin ^{2} x=17$

let $\sin ^{2} x=p$

$\frac{14}{p}-6 p=17 \Rightarrow 14-6 p^{2}=17 p$

$6 p^{2}+17 p-14=0$

$p=-3.5, \frac{2}{3} \Rightarrow \sin ^{2} x=\frac{2}{3}$

$\Rightarrow \sin x=\pm \sqrt{\frac{2}{3}}$

$	herefore$ Total $4$ solutions

The number of values of $x$ in the interval $\left(\frac{\pi}{4}, \frac{7 \pi}{4}\right)$ for which $14 \operatorname{cosec}^{2} x-2 \sin ^{2} x=21$ $-4 \cos ^{2} x$ holds, is

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