Need limit without using L'Hopital: $\lim_{x \to 0} \frac{\cot x \sin(\pi x)}{4\sec x}$

Question

Find $$\lim_{x \rightarrow 0} \dfrac{\cot x \sin(\pi x)}{4\sec x}$$

Thus far I have:

$$=\frac{1}{4} \lim_{x \rightarrow 0} \dfrac{\cos^2x \sin(\pi x)}{\sin x}$$

But where do I go from here? The answer is $\frac {\pi}{4}$ using L'Hopitals rule.

Answer 1

$$\frac{\cot x\sin\pi x}{4\sec x}=\frac14\frac{\frac{\cos x}{\sin x}\sin\pi x}{\frac1{\cos x}}=\frac14\cos^2x\frac{\sin\pi x}{\sin x}=$$

$$=\frac14\pi\cos^2x\;\frac{\sin\pi x}{\pi x}\frac x{\sin x}\xrightarrow[x\to 0]{}\frac14\pi\cdot 1\cdot 1\cdot 1=\frac\pi4 $$