To solve $\tan(-1) - \tan(3)$, we can apply the following principles:

1. Property of tangent for negative angles: $\tan(-x) = -\tan(x)$ Using this property: $\tan(-1) = -\tan(1)$

2. Now, the expression becomes: $-\tan(1) - \tan(3)$ This simplifies to: $-(\tan(1) + \tan(3))$

3. Now, let's calculate the values of $\tan(1)$ and $\tan(3)$. Using a calculator: $\tan(1) \approx 0.01746, \quad \tan(3) \approx 0.05241$

4. Substituting these values: $- (0.01746 + 0.05241) = -(0.06987)$

5. Therefore, the final result is approximately: $-0.06987$

Would you like more details on how to handle tangent values or any other specific parts?

Here are some related questions to expand on this:

1. What is the general behavior of trigonometric functions with negative angles?
2. How do we calculate tangent values for angles without a calculator?
3. What is the difference between radians and degrees in trigonometry?
4. How does the tangent function behave in different quadrants?
5. How do other trigonometric functions (sine, cosine) behave with negative angles?

Tip: Remember that tangent has a periodicity of $\pi$, meaning $\tan(x) = \tan(x + \pi)$.