Limit $\lim_{x\to 0}\frac{\sin\left(\pi\cos^2(x)\right)}{x^2}$

Question

Could someone help me solve this limit without L'Hopital's rule? $$ \lim_{x\to 0}\frac{\sin\left(\pi\cos^2(x)\right)}{x^2} $$

Answer 1

Use the identity $\sin(x)=\sin(\pi-x)$ and then multiply and divide by the argument of $\sin$.