# Trigonometry?

Geometry Level 4

$\cos A \sin^2 \left( \frac A2 \right) + \cos B \sin^2 \left( \frac B2 \right) + \cos C \sin^2 \left( \frac C2 \right)$

If $$A$$, $$B$$, and $$C$$ are the angles of a triangle, then the maximum value of the expression above can be written as $$\dfrac PQ$$, where $$P$$ and $$Q$$ are coprime positive integers, find $$Q-P$$.

Bonus: For what type of triangle will the value of this expression be maximum? Prove your result.

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