Given equation is $${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$

Put $${e^{{\mathop{\rm sinx}\nolimits} \,}} = t$$ in the given equation,

we get $${t^2} - 4t - 1 = 0$$

$$ \Rightarrow t = {{4 \pm \sqrt {16 + 4} } \over 2}$$

$$\,\,\,\,\,\,\,\,\,\,\, = {{4 \pm \sqrt {20} } \over 2}$$

$$\,\,\,\,\,\,\,\,\,\,\, = {{4 \pm 2\sqrt 5 } \over 2}$$

$$\,\,\,\,\,\,\,\,\,\,\, = 2 \pm \sqrt 5 $$

$$ \Rightarrow {e^{\sin x}} = 2 \pm \sqrt 5 $$ $$\,\,\,\,\,$$ (as $$t = {e^{\sin x}}$$)

$$ \Rightarrow {e^{\sin x}} = 2 - \sqrt 5 $$ and

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

$$ \Rightarrow {e^{\sin x}} = 2 - \sqrt 5 < 0$$

and $$\,\,\,\,\,\,\sin x = \ln \left( {2 + \sqrt 5 } \right) > 1$$ So, rejected

Hence given equation has no solution.

$$\therefore$$ The equation has no real roots.