Answer
$$0$$
Work Step by Step
Since $-1\leq \cos (\tan \theta)\leq 1$, then we have
$$-\cos \theta \leq\cos \theta \cos (\tan \theta)\leq\cos \theta.$$
Moreover, $\lim\limits_{\theta \to \frac{\pi}{2}} \cos\theta=\lim\limits_{\theta \to \frac{\pi}{2}}-\cos\theta=0$. Then by the Squeeze Theorem, we have
$$\lim\limits_{\theta \to \frac{\pi}{2}}\cos \theta \cos (\tan \theta)=0.$$
