Tricky trigonometry question

Geometry Level 3

The value of \( \cos \left( \dfrac{\text{arctan}(4/3)}2 \right) \) can be written as \(\pm \dfrac{a\sqrt b}c\), where \(a\), \(b\) and \(c\) are positive integers with \(b\) square-free and \(a\) and \(c\) coprime. Find \(a+b+c\).

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