Step 1: {Calculate the time to attain maximum altitude}
At maximum altitude, the projectile's vertical velocity component is zero. The time to reach this point is calculated as:
\[t = \frac{u \sin \theta}{g}\]
This equation can be rearranged to express \(u\) in terms of \(g\), \(t\), and \(\theta\):
\[u = \frac{g t}{\sin \theta}\]
Step 2: {Apply the horizontal range equation}
The horizontal range (\(R\)) of a projectile is given by:
\[R = u \cos \theta \times (2t)\]
Substitute the expression for \(u\) derived in Step 1:
\[R = \frac{g t}{\sin \theta} \cos \theta \times (2t)\]
Rearrange to isolate \(\cos \theta\):
\[\cos \theta = \frac{R}{2 u t}\]
Step 3: {Determine \( \cot \theta \)}
Using the derived relationships, we can express \(\cot \theta\) as:
\[\cot \theta = \frac{R}{2 g t^2}\]
Given that \(g = 10\):
\[\cot \theta = \frac{R}{2 \times 10 t^2} = \frac{R}{20 t^2}\]
Therefore, the correct option is (A) \( \frac{R}{20t^2} \).