Question

The number of real roots of the equation $e^{\sin x}-e^{-\sin x}-4=0$.

• A. $1$
• B. $2$
• C. Infinite
• D. None

Answer 1

The correct option is A $1$

given, $e^{\sin x}-e^{-\sin x}-4=0$

Let us consider.

$t=e^{\sin x}{e}^{-\sin x}=1/t$

$t-\frac{1}{t}=4$

$\frac{t^2-1}{t}=4$

$t^2-1=4t$

$t^2-4t-1=0$

$a=1b=-4c=-1$

$=\frac{+b\pm \sqrt{b^2-4ac}}{2a}$

$=\frac{+4\pm \sqrt{16+4}}{2}$

$=\frac{+4\pm 2\sqrt{5}}{2}$

$=\frac{2(+2\pm \sqrt{5})}{2}=+2\pm \sqrt{5}$

$e^{\sin }x=2\pm \sqrt{5}$

$-1\le \sin x\le 1$