1. **Problem Statement:** Prove that $$\sec^2\theta + \tan^2\theta = 2\tan^2\theta + 1$$. 2. **Recall the Pythagorean identity:** $$\sec^2\theta - \tan^2\theta = 1$$. 3. **Start with the left-hand side (LHS):** $$\sec^2\theta + \tan^2\theta$$ 4. **Rewrite $$\sec^2\theta$$ using the identity:** $$\sec^2\theta = 1 + \tan^2\theta$$ 5. **Substitute into LHS:** $$1 + \tan^2\theta + \tan^2\theta = 1 + 2\tan^2\theta$$ 6. **This matches the right-hand side (RHS):** $$2\tan^2\theta + 1$$ 7. **Therefore, the identity is proved:** $$\sec^2\theta + \tan^2\theta = 2\tan^2\theta + 1$$ This completes the proof in a clear, step-by-step manner.