Correct option is (4) 0
Take \(e^{\sin x}=t(t>0)\)
\(\Rightarrow \mathrm{t}-\frac{2}{\mathrm{t}}=2\)
\(\Rightarrow \frac{\mathrm{t}^{2}-2}{\mathrm{t}}=2\)
\(\Rightarrow \mathrm{t}^{2}-2 \mathrm{t}-2=0\)
\(\Rightarrow \mathrm{t}^{2}-2 \mathrm{t}+1=3\)
\(\Rightarrow(\mathrm{t}-1)^{2}=3\)
\(\Rightarrow \mathrm{t}=1 \pm \sqrt{3}\)
\(\Rightarrow \mathrm{t}=1 \pm 1.73\)
\(\Rightarrow \mathrm{t}=2.73 \text{ or }-0.73 \) (rejected as \(\mathrm{t}>0\))
\(\Rightarrow \mathrm{e}^{\sin \mathrm{x}}=2.73\)
\(\Rightarrow \log _{\mathrm{e}} \mathrm{e}^{\sin \mathrm{x}}=\log _{\mathrm{e}} 2.73\)
\(\Rightarrow \sin \mathrm{x}=\log _{\mathrm{e}} 2.73>1\)
So no solution.
