Chocolate mania (mechanics +trigonometry)

Cody is this time throwing a chocolate from the ground and the lady\color{#D61F06}{\text{lady}} is on the top of the building to catch it.

The point from where Cody throws the chocolate is 180m180\, \text{m} away from the building and building is 360m360\,\text{m} tall.

Given that when the lady\color{#D61F06}{\text{lady}} caught the chocolate, it was travelling horizontally (vertical velocity 0m/s0\,\text{m/s}) and Cody had thrown the chocolate with the initial velocity making an angle θ\theta with the horizontal, then find the value of

tanθ×tan(60θ)×tan(60+θ)\tan \theta \times \tan (60^\circ -\theta) \times \tan(60^\circ +\theta)

This value can be written as ab\dfrac{a}{b} for coprime positive integers aa and bb . then find the value of a+ba+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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