The number of values of x in the interval π 4 , 7 π 4 for which 14 cosec 2 x - 2 sin 2 x = 21 - 4…

Given $x\in \left(\frac{\pi }{4},\frac{7\pi }{4}\right)$

$14{cosec}^{2}x-2{\sin }^{2}x=21-4{\cos }^{2}x$

$=21-4\left(1-{\sin }^{2}x\right)$

$=17+4{\sin }^{2}x$

$\frac{14}{{\sin }^{2}x}-6{\sin }^{2}x=17$

Let ${\sin }^{2}x=p$

$\frac{14}{p}-6p=17\Rightarrow 14-6p^2=17p$

$6p^2+17p-14=0$

$p=-\frac{7}{2},\frac{2}{3}\Rightarrow {\sin }^{2}x=\frac{2}{3}$

$\Rightarrow \sin x=\pm \sqrt{\frac{2}{3}}$

$\therefore$  There are $4$ solutions