Find the value of `tan^(-1)(tan'(2pi)/(3))`.

Explanation

To find the value of \( \tan^{-1}(\tan(2\pi/3)) \), we can follow these steps:

### Step 1: Identify the angle
The angle given is \( \frac{2\pi}{3} \). We know that \( \tan^{-1}(\tan(x)) \) returns \( x \) if \( x \) is in the range of \( \tan^{-1} \), which is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

### Step 2: Determine the quadrant
The angle \( \frac{2\pi}{3} \) is in the second quadrant. In the second quadrant, the tangent function is negative.

### Step 3: Find the reference angle
The reference angle for \( \frac{2\pi}{3} \) can be found by subtracting it from \( \pi \):
\[ \text{Reference angle} = \pi - \frac{2\pi}{3} = \frac{\pi}{3} \]

### Step 4: Use the property of tangent
Since \( \tan\left(\frac{2\pi}{3}\right) = -\tan\left(\frac{\pi}{3}\right) \), we can write:
\[ \tan\left(\frac{2\pi}{3}\right) = -\sqrt{3} \]

### Step 5: Apply the inverse tangent function
Now, we can express \( \tan^{-1}(\tan(2\pi/3)) \) as:
\[ \tan^{-1}(\tan(2\pi/3)) = \tan^{-1}(-\sqrt{3}) \]

### Step 6: Find the angle corresponding to the negative tangent
The angle whose tangent is \( -\sqrt{3} \) in the range of \( \tan^{-1} \) is:
\[ -\frac{\pi}{3} \]

### Step 7: Conclusion
Thus, we have:
\[ \tan^{-1}(\tan(2\pi/3)) = -\frac{\pi}{3} \]

### Final Answer
The value of \( \tan^{-1}(\tan(2\pi/3)) \) is:
\[ -\frac{\pi}{3} \]

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