Product of Sine and Cosine

Geometry Level 1

\(\sin\left(\frac{\pi}{8}\right)\cos\left(\frac{\pi}{8}\right) = \frac{a\sqrt{b}}{c}\), where \(a, b\) and \(c\) are integers such that \(a\) and \(c\) are coprime, and \(b\) is not divisible by the square of any prime. What is the value of \(a+b+c\)?

Details and assumptions

\(a, b\) and \(c\) are allowed to be 1. In particular, if you think the value is \( 1 = \frac {1 \sqrt{1} } {1}\), then \(a+b+c=1+1+1=3\).

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