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Given:

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

$⇒ \dfrac{{x}^{2}}{5} - \dfrac{{y}^{2}}{5\sin^{2}\theta} = 1$ 

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

$⇒ \dfrac{{x}^{2}}{5\sin^{2}\theta} + \dfrac{{y}^{2}}{5} = 1$ \left\{\text{here}a < b\text{as}\sin^{2}\theta \le 1\right\}$</p><p>So,${e}_{H} = \sqrt{1 + \sin^{2}\theta} \text{ and } {e}_{E} = \sqrt{1 - \sin^{2}\theta}$</p><p>Also given,${e}_{H} = \sqrt{7}{e}_{E}$</p><p>$⇒ \sqrt{1 + \sin^{2}\theta} = \sqrt{7}\sqrt{1 - \sin^{2}\theta}$</p><p>$⇒ 1 + \sin^{2}\theta = 7 - 7\sin^{2}\theta$</p><p>$⇒ 8\sin^{2}\theta = 6$</p><p>$⇒ sin\theta = \dfrac{\sqrt{3}}{2}$</p><p>$⇒ \theta = \dfrac{\pi}{3}$