What happened to \(15^{\circ}\)?

Geometry Level 4

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.

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