Given that the terminal side of an angle $\alpha$ has a point $P(1, \sqrt{3})$, find the value of $\cos \alpha \cdot \tan \alpha$.

1. Problem Description

Given that the terminal side of an angle $\alpha$ has a point $P(1, \sqrt{3})$, find the value of $\cos \alpha \cdot \tan \alpha$.

2. Solution Steps

Let the point on the terminal side of angle $\alpha$ be $P(x, y)$, where $x = 1$ and $y = \sqrt{3}$.

We can find $r$, the distance from the origin to the point $P$, using the formula $r = \sqrt{x^2 + y^2}$.

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

Now, we can find $\cos \alpha$ and $\tan \alpha$ using the following formulas:

$\cos \alpha = \frac{x}{r}$

$\tan \alpha = \frac{y}{x}$

Thus, $\cos \alpha = \frac{1}{2}$ and $\tan \alpha = \frac{\sqrt{3}}{1} = \sqrt{3}$.

Now, we need to find the value of $\cos \alpha \cdot \tan \alpha$.

$\cos \alpha \cdot \tan \alpha = \frac{1}{2} \cdot \sqrt{3} = \frac{\sqrt{3}}{2}$

3. Final Answer

$\frac{\sqrt{3}}{2}$