The value of $\cot (\cos^{-1}\frac{7}{25})$ is

$\frac{7}{24}$

The correct answer is Option (4) → $\frac{7}{24}$

Let $\theta = \cos^{-1}\left(\frac{7}{25}\right)$

Then $\cos\theta = \frac{7}{25}$

$\sin\theta = \sqrt{1 - \cos^2\theta} = \sqrt{1 - \left(\frac{7}{25}\right)^2} = \sqrt{\frac{625 - 49}{625}} = \frac{24}{25}$

$\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{\frac{7}{25}}{\frac{24}{25}} = \frac{7}{24}$

Therefore, $\cot(\cos^{-1}\frac{7}{25}) = \frac{7}{24}$.