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Find $\cfrac{dy}{dx}$ at the point $\left( \pi, \cfrac{\pi}{4} \right)$ when $\sin x+ \sin y=\tan x+\cfrac{\sqrt{2}}{2}$.

A

$-2 \sqrt{2}$

B

$0$

C

$\cfrac{\sqrt{2}}{2}$

D

$2\sqrt{2}$

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