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Given, $$\operatorname{cosec}^2x-2=0$$
$$\Rightarrow$$ $$\operatorname{cosec}^2x=2$$
$$\Rightarrow$$ $$\operatorname{cosec}x=\sqrt{2}$$
$$\Rightarrow$$ $$\frac{1}{\sin x}=\sqrt{2}$$
$$\Rightarrow$$ $$\sin x=\frac{1}{\sqrt{2}}$$
Since $$0 < x < 90^\circ$$
$$\sin x=\sin45^{\circ\ }$$
$$\Rightarrow$$ $$x=45^{\circ\ }$$
Hence, the correct answer is Option D
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