A) 1
B) 2
C) Infinite
D) None
Correct Answer: D
Solution :
Given equation \[{{e}^{\sin x}}-{{e}^{-\sin x}}-4=0\] Let \[{{e}^{\sin x}}=y\], then given equation can be written as \[{{y}^{2}}-4y-1=0\]Þ \[y=2\pm \sqrt{5}\] But the value of \[y={{e}^{\sin x}}\] is always positive, so \[y=2+\sqrt{5}\,\,\,(\because 2<\sqrt{5})\] Þ \[{{\log }_{e}}y={{\log }_{e}}(2+\sqrt{5})\]Þ\[\sin x={{\log }_{e}}(2+\sqrt{5})>1\] which is impossible, since \[\sin x\] cannot be greater than 1. Hence we cannot find any real value of x which satisfies the given equation.
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