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

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

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