1. **State the problem:** Prove the trigonometric identity $$\sin^2 \theta + \cos^2 \theta + \tan^2 \theta = \sec^2 \theta$$. 2. **Recall fundamental identities:** - Pythagorean identity: $$\sin^2 \theta + \cos^2 \theta = 1$$. - Definition of tangent: $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$. - Definition of secant: $$\sec \theta = \frac{1}{\cos \theta}$$. 3. **Rewrite the left side using definitions:** $$\sin^2 \theta + \cos^2 \theta + \tan^2 \theta = 1 + \tan^2 \theta$$ (using the Pythagorean identity). 4. **Express $$\tan^2 \theta$$ in terms of sine and cosine:** $$1 + \left(\frac{\sin \theta}{\cos \theta}\right)^2 = 1 + \frac{\sin^2 \theta}{\cos^2 \theta}$$. 5. **Combine into a single fraction:** $$1 + \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta}{\cos^2 \theta} + \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta}$$. 6. **Use the Pythagorean identity again:** $$\frac{1}{\cos^2 \theta}$$. 7. **Recognize the right side:** $$\frac{1}{\cos^2 \theta} = \sec^2 \theta$$. 8. **Conclusion:** $$\sin^2 \theta + \cos^2 \theta + \tan^2 \theta = \sec^2 \theta$$ is proven true. This completes the proof.