Find the value of `tan { cot ^(-1) ((-2)/3)}`

Options

A. `- 2/3`
B. ` 2/3`
C. ` 3/2`
D. `- 3/2`

Answer

To find the value of \( \tan \left( \cot^{-1} \left( \frac{-2}{3} \right) \right) \), we can follow these steps:

### Step 1: Use the property of inverse cotangent
We know that:
\[ \cot^{-1}(-\theta) = \pi - \cot^{-1}(\theta) \]
Applying this property, we can rewrite our expression:
\[ \cot^{-1} \left( \frac{-2}{3} \right) = \pi - \cot^{-1} \left( \frac{2}{3} \right) \]

### Step 2: Substitute into the tangent function
Now, substituting this back into our expression, we have:
\[ \tan \left( \cot^{-1} \left( \frac{-2}{3} \right) \right) = \tan \left( \pi - \cot^{-1} \left( \frac{2}{3} \right) \right) \]

### Step 3: Use the tangent property
Using the property of tangent:
\[ \tan(\pi - \theta) = -\tan(\theta) \]
We can rewrite our expression as:
\[ \tan \left( \pi - \cot^{-1} \left( \frac{2}{3} \right) \right) = -\tan \left( \cot^{-1} \left( \frac{2}{3} \right) \right) \]

### Step 4: Find the tangent of the cotangent inverse
Next, we know that:
\[ \tan(\cot^{-1}(x)) = \frac{1}{x} \]
Thus, we can substitute \( x = \frac{2}{3} \):
\[ \tan \left( \cot^{-1} \left( \frac{2}{3} \right) \right) = \frac{1}{\frac{2}{3}} = \frac{3}{2} \]

### Step 5: Substitute back to find the final value
Now, substituting this back into our expression:
\[ \tan \left( \cot^{-1} \left( \frac{-2}{3} \right) \right) = -\frac{3}{2} \]

### Final Answer
Thus, the value of \( \tan \left( \cot^{-1} \left( \frac{-2}{3} \right) \right) \) is:
\[ \boxed{-\frac{3}{2}} \]