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Calculus Level 4

$\large {y=\sqrt{(x-sinx)+\sqrt{(x-sinx)+\ldots }}}$

If $$y$$ satisfy the equation above, then find

$${\left| |\frac{dx}{dy}|^{2}_{at~x=\frac{\pi}{2}}-4\displaystyle \lim_{n\to \infty}[\frac{1}{\sqrt{n^{2}-1}}+\frac{1}{\sqrt{n^{2}-2^{2}}}+....+\frac{1}{\sqrt{n^{2}-(n-1)^{2}}}] \right|}$$

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