The problem states that theta = (4)/(5).

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 $\sec \theta = \frac{5}{4}$.

Given $\sec \theta = \frac{5}{4}$.

$$\cos \theta = \frac{1}{\sec \theta}$$ $$\cos \theta = \frac{1}{5/4}$$ $$\cos \theta = \frac{4}{5}$$

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.

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 $\pm \frac{4}{3}$.