Answer
No discontinuities.
Work Step by Step
Discontinuities exist in $f(x)$ when the denominator of $f(x)$ equals $0$. Therefore, for a discontinuity to exist, $1+sin^2(x) = 0$.
$$1+sin^2(x) = 0$$
$$sin^2(x) = -1$$
$$sin(x) = i$$
There exist no $x$ where $sin(x) = i$. Therefore, there are no discontinuities.
