What happened to $15^{\circ}$?

The expression $\left(\dfrac{1}{\tan 5 + \cot 5}\right)\left(\dfrac{1}{\tan 25 + \cot 25}\right)\left(\dfrac{1}{\tan 35 + \cot 35}\right)\left(\dfrac{1}{\tan 45 + \cot 45}\right)$ can be expressed as $\dfrac{p}{q}$ for positive coprime integers $p,q$. Find $p+q$.

$\text{Details and Assumptions}$

All angles are in degrees.