Chocolate mania (mechanics +trigonometry)

Cody is this time throwing a chocolate from the ground and the \(\color{Red}{\text{lady}}\) is on the top of the building to catch it.

The point from where Cody throws the chocolate is \(180\, \text{m}\) away from the building and building is \(360\,\text{m}\) tall.

Given that when the \(\color{Red}{\text{lady}}\) caught the chocolate, it was travelling horizontally (vertical velocity \(0\,\text{m/s}\)) and Cody had thrown the chocolate with the initial velocity making an angle \(\theta\) with the horizontal, then find the value of

\[\tan \theta \times \tan (60^\circ -\theta) \times \tan(60^\circ +\theta)\]

This value can be written as \(\dfrac{a}{b}\) for coprime positive integers \(a\) and \(b\) . then find the value of \(a+b\).

Details: In this problem, you don't need the initial velocity at all, neither the value of acceleration due to gravity.


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