1. **State the problem:** Prove or verify the identity $$\sec^6\theta + \tan^6\theta = 1 + 3 \tan^2\theta \sec^2\theta$$. 2. **Recall the fundamental identity:** We know that $$\sec^2\theta = 1 + \tan^2\theta$$. 3. **Express everything in terms of $$\tan\theta$$:** Since $$\sec^2\theta = 1 + \tan^2\theta$$, then $$\sec^6\theta = (\sec^2\theta)^3 = (1 + \tan^2\theta)^3$$. 4. **Rewrite the left side:** $$\sec^6\theta + \tan^6\theta = (1 + \tan^2\theta)^3 + \tan^6\theta$$. 5. **Expand $$(1 + \tan^2\theta)^3$$ using the binomial theorem:** $$ (1 + x)^3 = 1 + 3x + 3x^2 + x^3 $$ where $$x = \tan^2\theta$$. So, $$ (1 + \tan^2\theta)^3 = 1 + 3 \tan^2\theta + 3 \tan^4\theta + \tan^6\theta $$. 6. **Substitute back:** $$\sec^6\theta + \tan^6\theta = 1 + 3 \tan^2\theta + 3 \tan^4\theta + \tan^6\theta + \tan^6\theta = 1 + 3 \tan^2\theta + 3 \tan^4\theta + 2 \tan^6\theta$$. 7. **Rewrite the right side:** $$1 + 3 \tan^2\theta \sec^2\theta = 1 + 3 \tan^2\theta (1 + \tan^2\theta) = 1 + 3 \tan^2\theta + 3 \tan^4\theta$$. 8. **Compare both sides:** Left side: $$1 + 3 \tan^2\theta + 3 \tan^4\theta + 2 \tan^6\theta$$ Right side: $$1 + 3 \tan^2\theta + 3 \tan^4\theta$$ They are not equal unless $$2 \tan^6\theta = 0$$, which is only true if $$\tan\theta = 0$$. **Conclusion:** The given identity $$\sec^6\theta + \tan^6\theta = 1 + 3 \tan^2\theta \sec^2\theta$$ is **not true** for all $$\theta$$. **Final answer:** The identity does not hold in general.