Question

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 __________

Answer 1

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

$\therefore$ Total $4$ solutions