Find the value of tan `(13pi)/(12)`.

Explanation

To find the value of \( \tan\left(\frac{13\pi}{12}\right) \), we can follow these steps:

### Step 1: Rewrite the angle
We start by rewriting \( \frac{13\pi}{12} \) in a more manageable form:
\[ \frac{13\pi}{12} = \pi + \frac{\pi}{12} \]
This shows that the angle is in the third quadrant, where the tangent function is positive.

### Step 2: Use the tangent addition formula
Using the tangent addition formula, we have:
\[ \tan(\pi + \theta) = \tan(\theta) \]
Thus, we can simplify:
\[ \tan\left(\frac{13\pi}{12}\right) = \tan\left(\frac{\pi}{12}\right) \]

### Step 3: Express \( \frac{\pi}{12} \) in terms of known angles
Next, we express \( \frac{\pi}{12} \) as the difference of two angles we know:
\[ \frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4} \]
Now we can apply the tangent subtraction formula:
\[ \tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \]
where \( x = \frac{\pi}{3} \) and \( y = \frac{\pi}{4} \).

### Step 4: Find the tangent values
We know:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \quad \text{and} \quad \tan\left(\frac{\pi}{4}\right) = 1 \]

### Step 5: Substitute into the formula
Now substituting these values into the tangent subtraction formula:
\[ \tan\left(\frac{\pi}{12}\right) = \frac{\tan\left(\frac{\pi}{3}\right) - \tan\left(\frac{\pi}{4}\right)}{1 + \tan\left(\frac{\pi}{3}\right) \tan\left(\frac{\pi}{4}\right)} \]
Substituting the known values:
\[ \tan\left(\frac{\pi}{12}\right) = \frac{\sqrt{3} - 1}{1 + \sqrt{3} \cdot 1} = \frac{\sqrt{3} - 1}{1 + \sqrt{3}} \]

### Step 6: Rationalize the denominator
To simplify further, we can rationalize the denominator:
\[ \tan\left(\frac{\pi}{12}\right) = \frac{(\sqrt{3} - 1)(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} \]
Calculating the denominator:
\[ (1 + \sqrt{3})(1 - \sqrt{3}) = 1 - 3 = -2 \]
Calculating the numerator:
\[ (\sqrt{3} - 1)(1 - \sqrt{3}) = \sqrt{3} - 3 - 1 + \sqrt{3} = 2\sqrt{3} - 4 \]
Thus, we have:
\[ \tan\left(\frac{\pi}{12}\right) = \frac{2\sqrt{3} - 4}{-2} = 2 - \sqrt{3} \]

### Final Result
Therefore, the value of \( \tan\left(\frac{13\pi}{12}\right) \) is:
\[ \tan\left(\frac{13\pi}{12}\right) = 2 - \sqrt{3} \]