Step 1: Recall standard trigonometric values
From the trigonometric table:
\[
\cos 0^\circ = 1,\ \cos 30^\circ = \frac{\sqrt{3}}{2},\ \cos 45^\circ = \frac{1}{\sqrt{2}},\ \cos 60^\circ = \frac{1}{2},\ \cos 90^\circ = 0
\]
Step 2: Apply for $\cos 60^\circ$
Clearly, \[
\cos 60^\circ = \frac{1}{2}
\]
\[
\boxed{\cos 60^\circ = \tfrac{1}{2}}
\]
The value of $\dfrac{1+\cot^2 \theta}{1+\tan^2 \theta}$ will be: