Answer
$$\lim\limits _{t \rightarrow 0} \sin \left(\frac{\pi}{2} \cos (\tan t)\right)=1
$$
The function is continuous at $ t=0$.
Work Step by Step
Given $$\lim\limits _{t \rightarrow 0} \sin \left(\frac{\pi}{2} \cos (\tan t)\right)
$$
So,
\begin{aligned}a)\lim\limits _{t \rightarrow 0} \sin \left(\frac{\pi}{2} \cos (\tan t)\right)&= \sin \left(\frac{\pi}{2} \cos (\tan 0)\right)\\
&= \sin \left(\frac{\pi}{2} \cos 0\right)\\
&= \sin \left(\frac{\pi}{2}\right)\\
&=1\\
\end{aligned}
Since $$ f(t)= \sin \left(\frac{\pi}{2} \cos (\tan t)\right)$$
\begin{aligned}b) f(0)&= \sin \left(\frac{\pi}{2} \cos (\tan 0)\right)\\
&= \sin \left(\frac{\pi}{2} \cos 0\right)\\
&= \sin \left(\frac{\pi}{2}\right)\\
&=1
\end{aligned}
From (a), (b) since $\lim \limits_{t \rightarrow 0} f(t)=f(0)=1,$ the function is continuous at $ t=0$.
