1. **Stating the problem:** We want to prove the trigonometric identity $$1 + \tan^2\alpha = \frac{1}{\cos^2\alpha}$$. 2. **Recall the definitions and formulas:** - By definition, $$\tan\alpha = \frac{\sin\alpha}{\cos\alpha}$$. - The Pythagorean identity states $$\sin^2\alpha + \cos^2\alpha = 1$$. 3. **Rewrite the left side using the definition of tangent:** $$1 + \tan^2\alpha = 1 + \left(\frac{\sin\alpha}{\cos\alpha}\right)^2 = 1 + \frac{\sin^2\alpha}{\cos^2\alpha}$$ 4. **Combine the terms over a common denominator:** $$1 + \frac{\sin^2\alpha}{\cos^2\alpha} = \frac{\cos^2\alpha}{\cos^2\alpha} + \frac{\sin^2\alpha}{\cos^2\alpha} = \frac{\cos^2\alpha + \sin^2\alpha}{\cos^2\alpha}$$ 5. **Use the Pythagorean identity:** $$\frac{\cos^2\alpha + \sin^2\alpha}{\cos^2\alpha} = \frac{1}{\cos^2\alpha}$$ 6. **Conclusion:** We have shown that $$1 + \tan^2\alpha = \frac{1}{\cos^2\alpha}$$, which proves the identity. This completes the proof.