Answer
$$\lim_{\theta\to0}\cos\Big(\frac{\pi\theta}{\sin\theta}\Big)=-1$$
Work Step by Step
$$A=\lim_{\theta\to0}\cos\Big(\frac{\pi\theta}{\sin\theta}\Big)$$
$$A=\cos\Big(\pi\lim_{\theta\to0}\frac{\theta}{\sin\theta}\Big)$$
Recall that $\lim_{x\to0}\frac{\sin x}{x}=1$ with $x=\theta$ here.
$$A=\cos(\pi\times1)$$
$$A=\cos\pi=-1$$
