The problem states that $\sec \theta = \frac{4}{5}$. Step 1: Determine the value of $\cos \theta$. We know that $\cos \theta = \frac{1}{\sec \theta}$. $$ \cos \theta = \frac{1}{4/5} = \frac{5}{4} $$ The value $\cos \theta = \frac{5}{4} = 1.25$ is outside the possible range for the cosine function, which is $[-1, 1]$. Therefore, no real angle $\theta$ exists for which $\sec \theta = \frac{4}{5}$. It is highly probable that there is a typo in the question, and it was intended to be $\sec \theta = \frac{5}{4}$ (or $\cos \theta = \frac{4}{5}$). We will proceed by assuming the intended value was $\sec \theta = \frac{5}{4}$. Assumption: We assume the question intended $\sec \theta = \frac{5}{4}$. Step 1: Find $\cos \theta$ using the reciprocal identity. Given $\sec \theta = \frac{5}{4}$. $$ \cos \theta = \frac{1}{\sec \theta} $$ $$ \cos \theta = \frac{1}{5/4} $$ $$ \cos \theta = \frac{4}{5} $$ Step 2: Find $\sin \theta$ using the Pythagorean identity. The Pythagorean identity is $\sin^2 \theta + \cos^2 \theta = 1$. Substitute the value of $\cos \theta$: $$ \sin^2 \theta + \left(\frac{4}{5}\right)^2 = 1 $$ $$ \sin^2 \theta + \frac{16}{25} = 1 $$ $$ \sin^2 \theta = 1 - \frac{16}{25} $$ $$ \sin^2 \theta = \frac{25}{25} - \frac{16}{25} $$ $$ \sin^2 \theta = \frac{9}{25} $$ Take the square root of both sides: $$ \sin \theta = \pm \sqrt{\frac{9}{25}} $$ $$ \sin \theta = \pm \frac{3}{5} $$ Since the quadrant of $\theta$ is not specified, $\sin \theta$ can be positive or negative. Step 3: Calculate $\cot \theta$. The identity for $\cot \theta$ is $\cot \theta = \frac{\cos \theta}{\sin \theta}$. Case 1: If $\sin \theta = \frac{3}{5}$ $$ \cot \theta = \frac{4/5}{3/5} $$ $$ \cot \theta = \frac{4}{3} $$ Case 2: If $\sin \theta = -\frac{3}{5}$ $$ \cot \theta = \frac{4/5}{-3/5} $$ $$ \cot \theta = -\frac{4}{3} $$ Therefore, based on the assumption that $\sec \theta = \frac{5}{4}$, there are two possible values for $\cot \theta$. The final answer is $\boxed{\pm \frac{4}{3}}$.