Answer
The function $h(x)=f(g(x))= \tan \frac{x}{2}$ is continuous everywhere except at the points$$\mathbb{R}- \{ (2n+1) \pi \mid n \in \mathbb{Z} \}.$$
Work Step by Step
As we know, the trigonometric functions $\sin x$ and $\cos x$ are continuous everywhere, so the function $h(x)=f(g(x))= \tan \frac{x}{2}=\frac{ \sin \frac{x}{2}}{\cos \frac{x}{2}}$ is continuous everywhere except at those points vanishing the denominator. So the discontinuities of the function are$$\cos \frac{x}{2}=0 \quad \Rightarrow \quad x= \{ (2n+1) \pi \mid n \in \mathbb{Z} \}$$Thus, the function $h(x)=f(g(x))= \tan \frac{x}{2}$ is continuous at the other points, $\mathbb{R}- \{ (2n+1) \pi \mid n \in \mathbb{Z} \}$, according to Theorem 1.12, Continuity of a Composite Function.
