The task is to find and rationalize the exact value of $\cot(15^\circ)$.

To solve this, we can use the cotangent identity:

$\cot(\theta) = \frac{1}{\tan(\theta)}$

Thus, we need to find $\tan(15^\circ)$ and then take the reciprocal. The tangent of $15^\circ$ can be found using a standard angle addition identity. We use the identity for $\tan(a + b)$:

$\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a) \tan(b)}$

We can break down $15^\circ$ as $45^\circ - 30^\circ$. Then we apply the tangent identity:

$\tan(15^\circ) = \tan(45^\circ - 30^\circ) = \frac{\tan(45^\circ) - \tan(30^\circ)}{1 + \tan(45^\circ) \tan(30^\circ)}$

Given that $\tan(45^\circ) = 1$ and $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, we substitute these values into the equation:

$\tan(15^\circ) = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}}$

Now simplify the numerator and denominator:

$\tan(15^\circ) = \frac{\frac{\sqrt{3} - 1}{\sqrt{3}}}{\frac{\sqrt{3} + 1}{\sqrt{3}}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}$

So, the exact value of $\tan(15^\circ)$ is $\frac{\sqrt{3} - 1}{\sqrt{3} + 1}$.

Next, to find $\cot(15^\circ)$, we take the reciprocal of $\tan(15^\circ)$:

$\cot(15^\circ) = \frac{1}{\tan(15^\circ)} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}$

Now, to rationalize the denominator, we multiply the numerator and denominator by $\sqrt{3} + 1$:

$\cot(15^\circ) = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3} - 1)(\sqrt{3} + 1)}$

Using the difference of squares for the denominator:

$(\sqrt{3} - 1)(\sqrt{3} + 1) = 3 - 1 = 2$

Now, expand the numerator:

$(\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}$

Thus, the rationalized form of $\cot(15^\circ)$ is:

$\cot(15^\circ) = \frac{4 + 2\sqrt{3}}{2}$

Simplifying the expression:

$\cot(15^\circ) = 2 + \sqrt{3}$

So, the exact rationalized value of $\cot(15^\circ)$ is:

$\boxed{2 + \sqrt{3}}$

Would you like further clarification on any of the steps, or would you like to explore more related problems? Here are some follow-up questions:

1. How does the tangent addition formula work for other angles?
2. Can you explain how to rationalize the denominator in trigonometric functions?
3. How can you apply the cotangent identity to different angles?
4. What other methods are available for calculating trigonometric values exactly?
5. How does understanding rationalization of trigonometric expressions help in simplifying complex problems?

Tip: When you need to rationalize an expression involving square roots, multiply the numerator and denominator by the conjugate of the denominator.