A) \[\frac{{{e}^{x}}\sin y+{{e}^{y}}\sin x}{{{e}^{y}}\cos x-{{e}^{x}}\cos y}\]
B) \[\frac{{{e}^{x}}\sin x+{{e}^{y}}\sin y}{{{e}^{y}}\cos x-{{e}^{x}}\cos y}\]
C) \[\frac{{{e}^{x}}\sin y-{{e}^{y}}\sin x}{{{e}^{y}}\cos x-{{e}^{x}}\cos y}\]
D) none of the above
Correct Answer: A
Solution :
\[{{e}^{x}}\sin y-{{e}^{y}}\cos x=1\] On differentiating both sides w.r.t. \[x,\] we get \[{{e}^{x}}\cos y\frac{dy}{dx}+{{e}^{x}}\sin y+{{e}^{y}}\sin x-\] \[\times \,\,{{e}^{y}}\cos x\frac{dy}{dx}=0\] \[\Rightarrow \] \[\frac{dy}{dx}({{e}^{x}}\cos y-{{e}^{y}}\cos x)\] \[=-({{e}^{x}}\sin y+{{e}^{y}}\sin x)\] \[\Rightarrow \] \[\frac{dy}{dx}=\frac{{{e}^{x}}\sin y+{{e}^{y}}\sin x}{({{e}^{y}}\cos x-{{e}^{x}}\cos y)}\]
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