1. **State the problem:** Prove the trigonometric identity $$\sec^4 x - 1 = 2 \tan^2 x + \tan^4 x.$$\n\n2. **Recall the fundamental identity:** $$\sec^2 x = 1 + \tan^2 x.$$ This is key to rewriting powers of secant in terms of tangent.\n\n3. **Rewrite the left side:**\n$$\sec^4 x - 1 = (\sec^2 x)^2 - 1 = (1 + \tan^2 x)^2 - 1.$$\n\n4. **Expand the square:**\n$$ (1 + \tan^2 x)^2 - 1 = (1 + 2 \tan^2 x + \tan^4 x) - 1 = 2 \tan^2 x + \tan^4 x.$$\n\n5. **Conclusion:** The left side simplifies exactly to the right side, so the identity holds true.\n\n**Final answer:** $$\sec^4 x - 1 = 2 \tan^2 x + \tan^4 x.$$