The value of `tan{(cos^(- 1)(-2/7)-pi/2)]` is

Explanation

To find the value of \( \tan\left(\cos^{-1}\left(-\frac{2}{7}\right) - \frac{\pi}{2}\right) \), we can follow these steps:

### Step 1: Rewrite the expression using properties of inverse trigonometric functions
We know that:
\[ \cos^{-1}(-x) = \pi - \cos^{-1}(x) \]
Thus, we can rewrite:
\[ \cos^{-1}\left(-\frac{2}{7}\right) = \pi - \cos^{-1}\left(\frac{2}{7}\right) \]
So, we have:
\[ \tan\left(\cos^{-1}\left(-\frac{2}{7}\right) - \frac{\pi}{2}\right) = \tan\left(\left(\pi - \cos^{-1}\left(\frac{2}{7}\right)\right) - \frac{\pi}{2}\right) \]

### Step 2: Simplify the expression
Using the identity \( \tan(\pi - x) = -\tan(x) \), we can simplify:
\[ \tan\left(\pi - \cos^{-1}\left(\frac{2}{7}\right) - \frac{\pi}{2}\right) = \tan\left(-\cos^{-1}\left(\frac{2}{7}\right)\right) = -\tan\left(\cos^{-1}\left(\frac{2}{7}\right)\right) \]

### Step 3: Find \( \tan(\cos^{-1}(x)) \)
We know that:
\[ \tan(\cos^{-1}(x)) = \frac{\sqrt{1-x^2}}{x} \]
For \( x = \frac{2}{7} \):
\[ \tan\left(\cos^{-1}\left(\frac{2}{7}\right)\right) = \frac{\sqrt{1 - \left(\frac{2}{7}\right)^2}}{\frac{2}{7}} \]

### Step 4: Calculate \( 1 - \left(\frac{2}{7}\right)^2 \)
Calculating:
\[ 1 - \left(\frac{2}{7}\right)^2 = 1 - \frac{4}{49} = \frac{49 - 4}{49} = \frac{45}{49} \]
Thus:
\[ \sqrt{1 - \left(\frac{2}{7}\right)^2} = \sqrt{\frac{45}{49}} = \frac{\sqrt{45}}{7} = \frac{3\sqrt{5}}{7} \]

### Step 5: Substitute back into the tangent formula
Now substituting back:
\[ \tan\left(\cos^{-1}\left(\frac{2}{7}\right)\right) = \frac{\frac{3\sqrt{5}}{7}}{\frac{2}{7}} = \frac{3\sqrt{5}}{2} \]

### Step 6: Final expression for \( \tan\left(\cos^{-1}\left(-\frac{2}{7}\right) - \frac{\pi}{2}\right) \)
Thus:
\[ \tan\left(\cos^{-1}\left(-\frac{2}{7}\right) - \frac{\pi}{2}\right) = -\tan\left(\cos^{-1}\left(\frac{2}{7}\right)\right) = -\frac{3\sqrt{5}}{2} \]

### Final Answer
The value of \( \tan\left(\cos^{-1}\left(-\frac{2}{7}\right) - \frac{\pi}{2}\right) \) is:
\[ -\frac{3\sqrt{5}}{2} \]