A) \[\frac{\sin y+{{e}^{y-x}}\sin x}{\cos y-{{e}^{y-x}}\cos x}\]
B) \[\frac{\cos y-{{e}^{y-x}}\cos x}{\sin y+{{e}^{y-x}}\sin x}\]
C) \[\frac{\cos y+{{e}^{y-x}}\cos x}{\sin y-{{e}^{y-x}}\sin x}\]
D) \[\frac{\sin y+{{e}^{y-x}}\sin x}{{{e}^{y-x}}\cos x-\cos y}\]
Correct Answer: D
Solution :
\[{{e}^{x}}\sin y-{{e}^{y}}\cos x=1\] On differentiating w.r.t.\[x,\] \[{{e}^{x}}\cos y.\frac{dy}{dx}+\sin y{{e}^{x}}-{{e}^{y}}\cos x\frac{dy}{dx}\] \[+{{e}^{y}}\sin x=0\] \[\Rightarrow \] \[\frac{dy}{dx}=-\frac{({{e}^{y}}\sin x+{{e}^{x}}\sin y)}{({{e}^{x}}\cos y-{{e}^{y}}\cos x)}\] \[\Rightarrow \] \[\frac{dy}{dx}=\frac{{{e}^{y}}\sin x+{{e}^{x}}\sin y}{{{e}^{x}}\cos x-{{e}^{y}}\cos y}\] \[\Rightarrow \] \[\frac{dy}{dx}=\frac{{{e}^{x}}({{e}^{y-x}}\sin x+\sin y)}{{{e}^{x}}({{e}^{y-x}}\cos x-\cos y)}\] \[\Rightarrow \] \[\frac{dy}{dx}=\frac{{{e}^{y-x}}\sin x+\sin y}{{{e}^{y-x}}\cos x-\cos y}\]
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