1. **State the problem:** Simplify the expression $$\cos^2 m (\tan m - 1)(\tan m + 1) - 2 \tan^2 m + \sec^2 m$$. 2. **Recall formulas and identities:** - $$\tan^2 m + 1 = \sec^2 m$$ - $$\cos^2 m = \frac{1}{\sec^2 m}$$ - Difference of squares: $$(a - b)(a + b) = a^2 - b^2$$ 3. **Simplify the product:** $$(\tan m - 1)(\tan m + 1) = \tan^2 m - 1$$ 4. **Substitute back:** $$\cos^2 m (\tan^2 m - 1) - 2 \tan^2 m + \sec^2 m$$ 5. **Rewrite using identities:** Since $$\cos^2 m = \frac{1}{\sec^2 m}$$, the expression becomes $$\frac{\tan^2 m - 1}{\sec^2 m} - 2 \tan^2 m + \sec^2 m$$ 6. **Express $$\tan^2 m$$ in terms of $$\sec^2 m$$:** From $$\tan^2 m + 1 = \sec^2 m$$, we get $$\tan^2 m = \sec^2 m - 1$$. 7. **Substitute $$\tan^2 m$$:** $$\frac{(\sec^2 m - 1) - 1}{\sec^2 m} - 2(\sec^2 m - 1) + \sec^2 m = \frac{\sec^2 m - 2}{\sec^2 m} - 2 \sec^2 m + 2 + \sec^2 m$$ 8. **Simplify each term:** $$\frac{\sec^2 m}{\sec^2 m} - \frac{2}{\sec^2 m} - 2 \sec^2 m + 2 + \sec^2 m = 1 - \frac{2}{\sec^2 m} - 2 \sec^2 m + 2 + \sec^2 m$$ 9. **Combine like terms:** $$1 + 2 + \sec^2 m - 2 \sec^2 m - \frac{2}{\sec^2 m} = 3 - \sec^2 m - \frac{2}{\sec^2 m}$$ 10. **Rewrite $$\frac{2}{\sec^2 m}$$ as $$2 \cos^2 m$$:** $$3 - \sec^2 m - 2 \cos^2 m$$ 11. **Final simplified expression:** $$\boxed{3 - \sec^2 m - 2 \cos^2 m}$$