The problem asks us to simplify the expression $\frac{\tan(\frac{\pi}{4}) + \tan(\frac{\pi}{6})}{1 - \tan(\frac{\pi}{4})\tan(\frac{\pi}{6})}$ and determine which of the given expressions is equivalent to it.

1. Problem Description

The problem asks us to simplify the expression $\frac{\tan(\frac{\pi}{4}) + \tan(\frac{\pi}{6})}{1 - \tan(\frac{\pi}{4})\tan(\frac{\pi}{6})}$ and determine which of the given expressions is equivalent to it.

2. Solution Steps

We can use the tangent addition formula:

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

Comparing this with the given expression $\frac{\tan(\frac{\pi}{4}) + \tan(\frac{\pi}{6})}{1 - \tan(\frac{\pi}{4})\tan(\frac{\pi}{6})}$, we can see that $a = \frac{\pi}{4}$ and $b = \frac{\pi}{6}$.

Thus, the expression simplifies to

$\tan(\frac{\pi}{4} + \frac{\pi}{6})$

To add the fractions $\frac{\pi}{4}$ and $\frac{\pi}{6}$, we need a common denominator. The least common multiple of 4 and 6 is

1

2. So, $\frac{\pi}{4} = \frac{3\pi}{12}$ and $\frac{\pi}{6} = \frac{2\pi}{12}$.

Therefore, $\frac{\pi}{4} + \frac{\pi}{6} = \frac{3\pi}{12} + \frac{2\pi}{12} = \frac{5\pi}{12}$.

So, the expression simplifies to $\tan(\frac{5\pi}{12})$.

3. Final Answer

$\tan(\frac{5\pi}{12})$