Is $ 1 - \sqrt{y^{2} + \cos^{2} \frac{\pi}{x}} < \frac{-2y^{2}x}{\sin \frac{\pi}{x} - 2xy}$ true for the following range of $y$?

Question

Suppose I have $y$ such that $$\frac{\sin \frac{\pi}{x}}{2x} < y < \sin\frac{\pi}{x}$$ with $x \geq 3$.

I cannot see a straightforward way to show that

$$ 1 - \sqrt{y^{2} + \cos^{2} \frac{\pi}{x}} < \frac{-2y^{2}x}{\sin \frac{\pi}{x} - 2xy}$$

?

Thanks!