Separate variables and integrate by parts

Near $x=0$, $\cos x\ne0$. Dividing and using $\sin2x=2\sin x\cos x$ gives $(e^y-ye^{-y})\,dy=2\sin x\,dx$.

For $\int-ye^{-y}\,dy$, take $u=y$, $dv=-e^{-y}dy$, so $du=dy$ and $v=e^{-y}$. Hence $\int-ye^{-y}\,dy=ye^{-y}-\int e^{-y}dy=ye^{-y}+e^{-y}$. Integrating both sides gives $e^y+ye^{-y}+e^{-y}=-2\cos x+C$.

At $x=0,y=0$, $2=-2+C$, so $C=4$. Thus $e^y+(y+1)e^{-y}=4-2\cos x$. Differentiating gives $(e^y-ye^{-y})y^\prime=2\sin x$; multiplying by $e^y\cos x$ recovers the original equation, including points where $\cos x=0$ when the implicit branch extends smoothly.