1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\sin(\pi \cos^2 x)}{x^2}.$$\n\n2. **Recall relevant formulas and rules:**\n- The limit $$\lim_{x \to 0} \frac{\sin x}{x} = 1.$$\n- For small angles, $$\sin y \approx y$$ when $$y \to 0.$$\n- We will use Taylor expansions to approximate $$\cos^2 x$$ near 0.\n\n3. **Analyze the inner function:**\n- $$\cos x \approx 1 - \frac{x^2}{2}$$ near 0.\n- Then $$\cos^2 x = (\cos x)^2 \approx \left(1 - \frac{x^2}{2}\right)^2 = 1 - x^2 + \frac{x^4}{4}.$$\n- So $$\pi \cos^2 x \approx \pi \left(1 - x^2 + \frac{x^4}{4}\right) = \pi - \pi x^2 + \frac{\pi x^4}{4}.$$\n\n4. **Rewrite the sine argument:**\n- $$\sin(\pi \cos^2 x) = \sin\left(\pi - \pi x^2 + \frac{\pi x^4}{4}\right).$$\n- Using the identity $$\sin(\pi - y) = \sin y,$$ we get\n $$\sin(\pi - \pi x^2 + \frac{\pi x^4}{4}) = \sin\left(\pi x^2 - \frac{\pi x^4}{4}\right).$$\n\n5. **Approximate sine for small arguments:**\n- For small $$z$$, $$\sin z \approx z.$$\n- So $$\sin\left(\pi x^2 - \frac{\pi x^4}{4}\right) \approx \pi x^2 - \frac{\pi x^4}{4}.$$\n\n6. **Form the original limit expression:**\n$$\frac{\sin(\pi \cos^2 x)}{x^2} \approx \frac{\pi x^2 - \frac{\pi x^4}{4}}{x^2} = \pi - \frac{\pi x^2}{4}.$$\n\n7. **Take the limit as $$x \to 0$$:**\n$$\lim_{x \to 0} \left(\pi - \frac{\pi x^2}{4}\right) = \pi.$$\n\n**Final answer:** $$\boxed{\pi}.$$