Find the value of `(tan 42^@ + tan 18^@)/(1-tan 42^@ tan 18^@)+ tan 300^@`

Answer

To solve the problem, we need to evaluate the expression:

\[ \frac{\tan 42^\circ + \tan 18^\circ}{1 - \tan 42^\circ \tan 18^\circ} + \tan 300^\circ \]

### Step 1: Use the tangent addition formula

The first part of the expression, \(\frac{\tan 42^\circ + \tan 18^\circ}{1 - \tan 42^\circ \tan 18^\circ}\), can be simplified using the tangent addition formula:

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

Here, let \(a = 42^\circ\) and \(b = 18^\circ\):

\[ \tan(42^\circ + 18^\circ) = \tan 60^\circ \]

### Step 2: Calculate \(\tan 60^\circ\)

We know that:

\[ \tan 60^\circ = \sqrt{3} \]

So, we can rewrite the first part of the expression:

\[ \frac{\tan 42^\circ + \tan 18^\circ}{1 - \tan 42^\circ \tan 18^\circ} = \tan 60^\circ = \sqrt{3} \]

### Step 3: Evaluate \(\tan 300^\circ\)

Next, we need to evaluate \(\tan 300^\circ\). We can express \(300^\circ\) in terms of a reference angle:

\[ 300^\circ = 360^\circ - 60^\circ \]

In the fourth quadrant, the tangent function is negative:

\[ \tan 300^\circ = -\tan 60^\circ = -\sqrt{3} \]

### Step 4: Combine the results

Now, we can combine the results from steps 2 and 3:

\[ \tan 60^\circ + \tan 300^\circ = \sqrt{3} - \sqrt{3} = 0 \]

### Final Result

Thus, the value of the expression is:

\[ \boxed{0} \]