Value of $tan\{cos^{-1}(-\frac{2}{7})-\frac{\pi}{2}\}$ is:

Question

Accepted Answer

To solve the problem and justify why Option 1 is correct, we need to evaluate the expression $ \tan\left(\cos^{-1}\left(-\frac{2}{7}\right) - \frac{\pi}{2}\right) $.

### Step-by-step Explanation:

1. **Understanding $\cos^{-1}\left(-\frac{2}{7}\right)$:**
   - The expression $\cos^{-1}\left(-\frac{2}{7}\right)$ represents an angle $\theta$ such that $\cos(\theta) = -\frac{2}{7}$.
   - In a right triangle, if $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$, then here, the adjacent side is $-2$ and the hypotenuse is $7$.

2. **Finding the opposite side:**
   - Using the Pythagorean theorem: $\text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2$.
   - $\text{opposite}^2 + (-2)^2 = 7^2$.
   - $\text{opposite}^2 + 4 = 49$.
   - $\text{opposite}^2 = 45$.
   - $\text{opposite} = \sqrt{45} = 3\sqrt{5}$.

3. **Finding $\tan(\theta)$:**
   - $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3\sqrt{5}}{-2} = -\frac{3\sqrt{5}}{2}$.

4. **Evaluating $\tan\left(\theta - \frac{\pi}{2}\right)$:**
   - Using the tangent subtraction formula: $\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}$.
   - Here, $a = \theta$ and $b = \frac{\pi}{2}$, so $\tan\left(\frac{\pi}{2}\right)$ is undefined, but we can use the identity $\tan\left(\theta - \frac{\pi}{2}\right) = -\cot(\theta)$.
   - $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{-2}{3\sqrt{5}}$.
   - Therefore, $\tan\left(\theta - \frac{\pi}{2}\right) = -\left(-\frac{2}{3\sqrt{5}}\right) = \frac{2}{3\sqrt{5}}$.

### Conclusion:

- The value of $\tan\left(\cos^{-1}\left(-\frac{2}{7}\right) - \frac{\pi}{2}\right)$ is indeed $\frac{2}{3\sqrt{5}}$, which matches Option 1.

### Explanation of Incorrect Options:

- **Option 2: $\frac{2}{3}$**
  - This option does not account for the $\sqrt{5}$ factor in the denominator, which arises from the Pythagorean calculation of the opposite side.

- **Option 3: $\frac{1}{\sqrt{5}}$**
  - This option does not match the calculated value of $\tan\left(\theta - \frac{\pi}{2}\right)$ and ignores the factor of $\frac{2}{3}$.

- **Option 4: $\frac{4}{5}$**
  - This option is unrelated to the calculated tangent value and does not consider the specific trigonometric identities and calculations involved.

Thus, Option 1 is the correct answer, as it accurately reflects the calculated value of the given trigonometric expression.