1. **State the problem:** Verify the trigonometric identity $$\sec^6 \theta + \tan^6 \theta = 1 + 3 \tan^2 \theta \sec^2 \theta$$. 2. **Recall key identities:** - $$\sec^2 \theta = 1 + \tan^2 \theta$$ - This identity will help us express everything in terms of $$\tan \theta$$. 3. **Rewrite the left side:** $$\sec^6 \theta + \tan^6 \theta = (\sec^2 \theta)^3 + (\tan^2 \theta)^3$$ 4. **Use the sum of cubes formula:** $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$ where $$a = \sec^2 \theta$$ and $$b = \tan^2 \theta$$. 5. **Apply the formula:** $$= (\sec^2 \theta + \tan^2 \theta)(\sec^4 \theta - \sec^2 \theta \tan^2 \theta + \tan^4 \theta)$$ 6. **Substitute $$\sec^2 \theta = 1 + \tan^2 \theta$$:** $$\sec^2 \theta + \tan^2 \theta = (1 + \tan^2 \theta) + \tan^2 \theta = 1 + 2 \tan^2 \theta$$ 7. **Simplify the second factor:** - Calculate each term: - $$\sec^4 \theta = (\sec^2 \theta)^2 = (1 + \tan^2 \theta)^2 = 1 + 2 \tan^2 \theta + \tan^4 \theta$$ - $$- \sec^2 \theta \tan^2 \theta = - (1 + \tan^2 \theta) \tan^2 \theta = - \tan^2 \theta - \tan^4 \theta$$ - $$+ \tan^4 \theta$$ Sum these: $$1 + 2 \tan^2 \theta + \tan^4 \theta - \tan^2 \theta - \tan^4 \theta + \tan^4 \theta = 1 + (2 \tan^2 \theta - \tan^2 \theta) + (\tan^4 \theta - \tan^4 \theta + \tan^4 \theta) = 1 + \tan^2 \theta + \tan^4 \theta$$ 8. **Multiply the two factors:** $$(1 + 2 \tan^2 \theta)(1 + \tan^2 \theta + \tan^4 \theta)$$ 9. **Expand:** $$= 1(1 + \tan^2 \theta + \tan^4 \theta) + 2 \tan^2 \theta (1 + \tan^2 \theta + \tan^4 \theta)$$ $$= 1 + \tan^2 \theta + \tan^4 \theta + 2 \tan^2 \theta + 2 \tan^4 \theta + 2 \tan^6 \theta$$ 10. **Combine like terms:** $$= 1 + (\tan^2 \theta + 2 \tan^2 \theta) + (\tan^4 \theta + 2 \tan^4 \theta) + 2 \tan^6 \theta$$ $$= 1 + 3 \tan^2 \theta + 3 \tan^4 \theta + 2 \tan^6 \theta$$ 11. **Rewrite the right side of the original equation:** $$1 + 3 \tan^2 \theta \sec^2 \theta = 1 + 3 \tan^2 \theta (1 + \tan^2 \theta) = 1 + 3 \tan^2 \theta + 3 \tan^4 \theta$$ 12. **Compare both sides:** - Left side simplified to $$1 + 3 \tan^2 \theta + 3 \tan^4 \theta + 2 \tan^6 \theta$$ - Right side is $$1 + 3 \tan^2 \theta + 3 \tan^4 \theta$$ 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 in general**. **Final answer:** The identity does not hold for all $$\theta$$.