Find the exact value of the expression. \( \tan \left[\sin ^{-1}\left(\frac{4}{9}\right)\right] \) Select the correct choice and fill in any answer boxes in your choice below. A. \( \tan \left[\sin ^{-1}\left(\frac{4}{9}\right)\right]=\square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. There is no solution.

To solve for \( \tan \left[\sin ^{-1}\left(\frac{4}{9}\right)\right] \), we can set \( \theta = \sin^{-1}\left(\frac{4}{9}\right) \). This means that \( \sin(\theta) = \frac{4}{9} \). Using the Pythagorean identity, we can find \( \cos(\theta) \): \[ \cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(\frac{4}{9}\right)^2} = \sqrt{1 - \frac{16}{81}} = \sqrt{\frac{65}{81}} = \frac{\sqrt{65}}{9}. \] Now we can find \( \tan(\theta) \): \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{4}{9}}{\frac{\sqrt{65}}{9}} = \frac{4}{\sqrt{65}}. \] Thus, the exact value of the expression is: \[ \tan \left[\sin ^{-1}\left(\frac{4}{9}\right)\right] = \frac{4}{\sqrt{65}}. \] To simplify further, we can rationalize the denominator: \[ \tan \left[\sin ^{-1}\left(\frac{4}{9}\right)\right] = \frac{4}{\sqrt{65}} \cdot \frac{\sqrt{65}}{\sqrt{65}} = \frac{4\sqrt{65}}{65}. \] So the answer is: A. \( \tan \left[\sin ^{-1}\left(\frac{4}{9}\right)\right]=\frac{4\sqrt{65}}{65} \)