$AB$ is a vertical pole. The end $A$ is on the level ground $C$ is the mid point of $AB. P$ is a point on the level ground such that the portion $BC$ subtends an angle $\displaystyle \Theta $ at $P$. If $AP = nAB$ then the value of $\displaystyle \cot \Theta $ is

Let AB = h, so BC = h/2. Let A be at origin (0,0), B at (0,h), P at (d,0). AP = d = n*AB = nh. Angle at P subtended by BC is theta. Angle APC = alpha, Angle APB = beta. tan(alpha) = AC/AP = (h/2) / (nh) = 1/(2n). tan(beta) = AB/AP = h / (nh) = 1/n. theta = beta - alpha. tan(theta) = (1/n - 1/2n) / (1 + 1/2n^2) = (1/2n) / ((2n^2+1)/2n^2) = n / (2n^2+1). cot(theta) = (2n^2+1)/n.