A) one and only one solution
B) four solutions
C) infinite number of solutions
D) no solution
Correct Answer: D
Solution :
Given, \[{{e}^{\sin \,x}}-{{e}^{-\sin x}}=16\] Let \[{{e}^{\sin \,x}}=t\] \[\therefore \] \[t=\frac{1}{t}=16\] \[\Rightarrow \] \[{{t}^{2}}-16t-1=0\] \[\therefore \] \[t=\frac{16\pm \sqrt{{{(16)}^{2}}+4}}{2(1)}=\frac{16\pm \sqrt{260}}{2}\] \[\Rightarrow \] \[t=\frac{16\pm 2\sqrt{65}}{2}\] \[\Rightarrow \] \[t=8+\sqrt{65,}\,8-\sqrt{65}\] \[\therefore \] \[{{e}^{sin\,\,x}}=8+\sqrt{65},\,\,{{e}^{\sin \,x}}=8-\sqrt{65}\] \[\sin x={{\log }_{e}}(8+\sqrt{65})\] \[\sin x\ne {{\log }_{e}}(8-\sqrt{65})\] Here, \[{{\log }_{e}}\,(8+\sqrt{65})>1\] \[(\because \,\,\sqrt{65}>8)\] Which is not possible
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