1. **State the problem:** Simplify the expression $\frac{\cos \theta}{1 + \tan^2 \theta}$. 2. **Recall the Pythagorean identity:** We know that $1 + \tan^2 \theta = \sec^2 \theta$. 3. **Rewrite the expression using the identity:** $$\frac{\cos \theta}{1 + \tan^2 \theta} = \frac{\cos \theta}{\sec^2 \theta}$$ 4. **Express $\sec^2 \theta$ in terms of cosine:** $$\sec^2 \theta = \frac{1}{\cos^2 \theta}$$ 5. **Substitute back:** $$\frac{\cos \theta}{\frac{1}{\cos^2 \theta}} = \cos \theta \times \cos^2 \theta$$ 6. **Simplify multiplication:** $$\cos \theta \times \cos^2 \theta = \cos^3 \theta$$ 7. **Final answer:** $$\boxed{\cos^3 \theta}$$