Use an appropriate Pythagorean identity to find the indicated value for an acute angle \( \theta \). Write your answer as a simplified fraction denominators if necessary. Given \( \csc \theta=\frac{29}{20} \), find the exact value of \( \cot \theta \).

To find \( \cot \theta \) given \( \csc \theta = \frac{29}{20} \), we first recall that \( \csc \theta = \frac{1}{\sin \theta} \). This allows us to determine \( \sin \theta \): \[ \sin \theta = \frac{20}{29}. \] Next, we can find \( \cos \theta \) using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \): \[ \sin^2 \theta = \left(\frac{20}{29}\right)^2 = \frac{400}{841}. \] Now substituting into the identity: \[ \frac{400}{841} + \cos^2 \theta = 1, \] \[ \cos^2 \theta = 1 - \frac{400}{841} = \frac{841 - 400}{841} = \frac{441}{841}. \] Taking the square root gives: \[ \cos \theta = \frac{21}{29}. \] Finally, \( \cot \theta \) is the ratio of \( \cos \theta \) to \( \sin \theta \): \[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{21}{29}}{\frac{20}{29}} = \frac{21}{20}. \] Thus, the exact value of \( \cot \theta \) is \( \frac{21}{20} \).