1. **State the problem:** Prove or verify the identity $$ (1 + \tan^2 \theta) \cos^2 \theta = 1 $$. 2. **Recall the Pythagorean identity:** We know that $$ 1 + \tan^2 \theta = \sec^2 \theta $$. 3. **Substitute the identity:** Replace $$ 1 + \tan^2 \theta $$ with $$ \sec^2 \theta $$ in the original expression: $$ (1 + \tan^2 \theta) \cos^2 \theta = \sec^2 \theta \cos^2 \theta $$ 4. **Express secant in terms of cosine:** Recall that $$ \sec \theta = \frac{1}{\cos \theta} $$, so $$ \sec^2 \theta = \frac{1}{\cos^2 \theta} $$. 5. **Substitute and simplify:** $$ \sec^2 \theta \cos^2 \theta = \frac{1}{\cos^2 \theta} \cos^2 \theta $$ 6. **Cancel common factors:** $$ \frac{\cancel{\cos^2 \theta}}{\cancel{\cos^2 \theta}} = 1 $$ 7. **Conclusion:** The left side simplifies to 1, so the identity is verified: $$ (1 + \tan^2 \theta) \cos^2 \theta = 1 $$ This shows the original equation is true for all values of $$ \theta $$ where the expressions are defined.