`tan^(-1)(tan 5)=`

Options

A. 5
B. `1/5`
C. `2pi-5`
D. `5-2pi`

Answer

To solve the problem \( \tan^{-1}(\tan 5) \), we need to understand the properties of the tangent and its inverse function. The tangent function has a periodicity of \( \pi \), and the range of the inverse tangent function \( \tan^{-1}(x) \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

### Step-by-Step Solution:

1. **Identify the angle**: We start with the expression \( \tan^{-1}(\tan 5) \). The angle \( 5 \) is in radians.

2. **Determine the equivalent angle**: Since \( \tan \) is periodic with a period of \( \pi \), we need to find an equivalent angle of \( 5 \) that lies within the range of \( \tan^{-1} \). We can do this by subtracting \( \pi \) from \( 5 \) until the angle falls within \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

\[ 5 - \pi \approx 5 - 3.14 \approx 1.86 \]

Since \( 1.86 \) is still greater than \( \frac{\pi}{2} \) (approximately \( 1.57 \)), we subtract \( \pi \) again:

\[ 1.86 - \pi \approx 1.86 - 3.14 \approx -1.28 \]

Now, \( -1.28 \) lies within the range of \( \tan^{-1} \).

3. **Use the property of tangent**: We have established that:

\[ \tan(5) = \tan(-1.28) \]

Therefore, we can write:

\[ \tan^{-1}(\tan 5) = -1.28 \]

4. **Convert back to the original angle**: Since we want the value in terms of \( 5 \):

\[ \tan^{-1}(\tan 5) = 5 - \pi \]

5. **Final answer**: Thus, the final answer is:

\[ \tan^{-1}(\tan 5) = 5 - \pi \]

### Conclusion:
The value of \( \tan^{-1}(\tan 5) \) is \( 5 - \pi \).