For $0<\theta <\pi /2$, if the eccentricity of the hyperbola $x^2-y^2{\mathrm{cosec}}^2\theta =5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^2{\mathrm{cosec}}^2\theta +y^2=5$, then the value of $\theta$ is:

Given:

The equation of hyperbola is, $x^2-y^2{\mathrm{cosec}}^2\theta =5$

$\Rightarrow \frac{x^2}{5}-\frac{y^2}{5{\sin }^2\theta }=1$ 

And the equation of ellipse is $x^2{\mathrm{cosec}}^2\theta +y^2=5$

$\Rightarrow \frac{x^2}{5{\sin }^2\theta }+\frac{y^2}{5}=1$ $\left\{\text{here} a<b \text{as} {\sin }^2\theta \le 1\right\}$

So, $e_H=\sqrt{1+{\sin }^2\theta } \& e_E=\sqrt{1-{\sin }^2\theta }$

Also given, $e_H=\sqrt{7}{e}_E$

$\Rightarrow \sqrt{1+{\sin }^2\theta } =\sqrt{7}\sqrt{1-{\sin }^2\theta }$

$\Rightarrow 1+{\sin }^2\theta =7-7{\sin }^2\theta$

$\Rightarrow 8{\sin }^2\theta =6$

$\Rightarrow \sin \theta =\frac{\sqrt{3}}{2}$

$\Rightarrow \theta =\frac{\mathrm{\pi }}{3}$