Find the value of : `tan^(-1){tan(-(7pi)/8)}`

Explanation

To solve the problem \( \tan^{-1}(\tan(-\frac{7\pi}{8})) \), we need to follow a series of steps to find the correct value.

### Step-by-Step Solution:

1. **Understanding the Function**: We know that the function \( \tan^{-1}(x) \) (the inverse tangent function) has a range of \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \). This means that any input to \( \tan^{-1} \) must be adjusted to fall within this range.

2. **Rewrite the Angle**: The angle \( -\frac{7\pi}{8} \) is outside the range of \( \tan^{-1} \). To bring it into a more manageable form, we can use the periodic properties of the tangent function:
\[ -\frac{7\pi}{8} = \pi - \frac{7\pi}{8} - \pi = -\left(\pi - \frac{7\pi}{8}\right) = -\frac{\pi}{8} \]
This means we can rewrite:
\[ \tan(-\frac{7\pi}{8}) = \tan(-(\pi - \frac{7\pi}{8})) = \tan(-\frac{\pi}{8}) \]

3. **Using the Property of Tangent**: We know that:
\[ \tan(\pi - \theta) = -\tan(\theta) \]
Thus:
\[ \tan(-\frac{7\pi}{8}) = \tan(-\frac{\pi}{8}) = -\tan(\frac{\pi}{8}) \]

4. **Apply the Inverse Tangent**: Now we can substitute this back into our original expression:
\[ \tan^{-1}(\tan(-\frac{7\pi}{8})) = \tan^{-1}(-\tan(\frac{\pi}{8})) \]

5. **Using the Property of Inverse Tangent**: The property of the inverse tangent function states:
\[ \tan^{-1}(-x) = -\tan^{-1}(x) \]
Therefore:
\[ \tan^{-1}(-\tan(\frac{\pi}{8})) = -\tan^{-1}(\tan(\frac{\pi}{8})) \]

6. **Final Simplification**: Since \( \frac{\pi}{8} \) lies within the range of \( \tan^{-1} \), we can cancel the tangent and inverse tangent:
\[ -\tan^{-1}(\tan(\frac{\pi}{8})) = -\frac{\pi}{8} \]

Thus, the final answer is:
\[ \boxed{-\frac{\pi}{8}} \]