Given that \(\tan \alpha = 4\), we can calculate \(\cos \alpha\), \(\sin \alpha\), and \(\cot \alpha\).

1. **Find \(\sin \alpha\) and \(\cos \alpha\)**:
\[
\tan \alpha = \frac{\sin \alpha}{\cos \alpha} = 4
\]
Let \(\sin \alpha = 4k\) and \(\cos \alpha = k\) for some \(k\). Then:
\[
\tan \alpha = \frac{4k}{k} = 4
\]
We can use the Pythagorean identity:
\[
\sin^2 \alpha + \cos^2 \alpha = 1
\]
Substituting \(\sin \alpha\) and \(\cos \alpha\):
\[
(4k)^2 + k^2 = 1 \implies 16k^2 + k^2 = 1 \implies 17k^2 = 1 \implies k^2 = \frac{1}{17} \implies k = \frac{1}{\sqrt{17}}
\]
Therefore:
\[
\cos \alpha = k = \frac{1}{\sqrt{17}}, \quad \sin \alpha = 4k = \frac{4}{\sqrt{17}}
\]

2. **Find \(\cot \alpha\)**:
\[
\cot \alpha = \frac{1}{\tan \alpha} = \frac{1}{4}
\]

3. **Results**:
- \(\cos \alpha = \frac{1}{\sqrt{17}}\)
- \(\sin \alpha = \frac{4}{\sqrt{17}}\)
- \(\cot \alpha = \frac{1}{4}\)

These are the values of \(\cos \alpha\), \(\sin \alpha\), and \(\cot \alpha\).