Find all real numbers $x$ for which $\tan (x / 2)$ is defined and greater than $\sin (x)$.
We claim that the answer is $x \in\left(k \pi-\frac{\pi}{2}, k \pi\right)$ for $k \in \mathbb{Z}$.

The statement is never true for $x / 2$ a multiple of $\frac{\pi}{2}$ because $\tan x / 2$ is either undefined or equal to $0=\sin x$. Thus $x / 2$ is not a multiple of $\frac{\pi}{2}$, so $\cos ^{2}(x / 2)>0$, and

$$
\begin{aligned}
\tan (x / 2) & \stackrel{?}{>} \sin (x) \\
\cos ^{2}(x / 2) \tan (x / 2) & \stackrel{?}{>} \cos ^{2}(x / 2) \sin (x) \\
2 \cos (x / 2) \sin (x / 2) & \stackrel{?}{>} 2 \cos ^{2}(x / 2) \sin (x) \\
\sin (x) & \stackrel{?}{>}(1+\cos (x)) \sin (x) \\
0 & \stackrel{?}{>} \cos (x) \sin (x) \\
0 & \stackrel{?}{>} \sin (2 x),
\end{aligned}
$$

as desired.