A) 0
B) 1
C) 2
D) 4
Correct Answer: A
Solution :
The given equation can be written as \[{{e}^{\sin x}}=4+\frac{1}{{{e}^{\sin x}}}\] ?.(1) Now \[-1\le \sin x\le 1\] and \[e<3\] \[\Rightarrow \,\,{{e}^{\sin x}}<3\] \[\Rightarrow \] Again as we always have \[\frac{1}{{{e}^{\sin x}}}>0\] \[\therefore \,\,4+\frac{1}{{{e}^{\sin x}}}>4\] Thus the L.H.S of (1) \[<3\] and R.H.S of \[(1)>4.\] Hence there is no real values of x which satisfy (1). It follows that the given equation has no real solution.
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