The function $$f(x) = e^{\sin x} - e^{-\sin x} - 4$$ is always greater than 0, thus no real solution exists.

The function $$f(x) = e^{\sin x} - e^{-\sin x} - 4$$ is always less than 0, thus no real solution exists.

The function $$f(x) = e^{\sin x} - e^{-\sin x} - 4$$ has a minimum value of -4, thus no real solution exists.

The function $$f(x) = e^{\sin x} - e^{-\sin x} - 4$$ is oscillating between -5 and -3, thus no real solution exists.

The function $$f(x) = e^{\sin x} - e^{-\sin x} - 4$$ is unbounded, thus no real solution exists.