What happened to 1515^{\circ}?

Geometry Level 4

The expression (1tan5+cot5)(1tan25+cot25)(1tan35+cot35)(1tan45+cot45)\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 pq\dfrac{p}{q} for positive coprime integers p,qp,q. Find p+qp+q.

Details and Assumptions\text{Details and Assumptions}

All angles are in degrees.

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