Calculate the exact value of tan(15 degrees).

We can write $15^\circ$ as $45^\circ - 30^\circ$. We will use the tangent subtraction formula.

The formula for $\tan(A - B)$ is:

Substitute $A = 45^\circ$ and $B = 30^\circ$:

We know that $\tan(45^\circ) = 1$ and $\tan(30^\circ) = \frac{1}{\sqrt{3}}$.
Substitute these values into the expression:

To simplify, multiply the numerator and the denominator by $\sqrt{3}$:

Now, rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $(\sqrt{3} - 1)$:

Expand the numerator and denominator:
Numerator: $(\sqrt{3} - 1)^2 = (\sqrt{3})^2 - 2(\sqrt{3})(1) + (1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}$
Denominator: $(\sqrt{3} + 1)(\sqrt{3} - 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2$
Substitute these back into the expression:

Factor out 2 from the numerator:

Cancel out the 2 in the numerator and denominator:

Thus, we have shown that $\tan(15^\circ) = 2 - \sqrt{3}$.