Question

$If{e}^{sinx}-e^{-sinx}-4=0,then,x=$

• A. 0
• B. $si{n}^{-1}\left\{lo{g}_e\right(2-\sqrt{5}\left)\right\}$
• C. 1
• D. none of these

Answer 1

The correct option is D

none of these

$Givenequation:e^{sinx}-e^{-sinx}-4=0Let:e^{sinx}=yNow,y-y^{-1}-4=0\Rightarrow y^2-4y-1=0\therefore y=\frac{4\pm \sqrt{16+4}}{2}\Rightarrow y=\frac{4\pm \sqrt{20}}{2}\Rightarrow y=\frac{4\pm \sqrt{5}}{2}=2\pm \sqrt{5}And,y=e^{sin}x\Rightarrow e^{sin}x=2\pm \sqrt{5}Takinglogonbothsides,weget:sinx=lo{g}_e(2\pm \sqrt{5})\Rightarrow sinx=sinx=lo{g}_e(2\pm \sqrt{5})orsinx=lo{g}_e(2-\sqrt{5})\Rightarrow sinx=lo{g}_e\left(4.24\right)orsinx=lo{g}_e(-0.24)lo{g}_e\left(4.24\right)>1andsinxcannotbegreaterthan1.\text{in the other case, the log of negative term occurs, which is not defined.}$