1. **State the problem:** Simplify the expression $$\frac{2+\tan^2 x}{\sec^2 x} - 1 = g(x)$$. 2. **Recall the identity:** We know that $$\sec^2 x = 1 + \tan^2 x$$. 3. **Rewrite the numerator:** The numerator is $$2 + \tan^2 x$$. 4. **Substitute the identity in the denominator:** Replace $$\sec^2 x$$ with $$1 + \tan^2 x$$. 5. **Rewrite the expression:** $$g(x) = \frac{2 + \tan^2 x}{1 + \tan^2 x} - 1$$ 6. **Combine into a single fraction:** $$g(x) = \frac{2 + \tan^2 x}{1 + \tan^2 x} - \frac{1 + \tan^2 x}{1 + \tan^2 x} = \frac{2 + \tan^2 x - (1 + \tan^2 x)}{1 + \tan^2 x}$$ 7. **Simplify the numerator:** $$2 + \tan^2 x - 1 - \tan^2 x = 1$$ 8. **Final simplified form:** $$g(x) = \frac{1}{1 + \tan^2 x}$$ 9. **Use the identity again:** Since $$1 + \tan^2 x = \sec^2 x$$, $$g(x) = \frac{1}{\sec^2 x} = \cos^2 x$$. **Answer:** $$g(x) = \cos^2 x$$