Question

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

Answer 1

Answer: 4

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

$\therefore$ Total $4$ solutions