Trigonometric functions have got large powers!

Geometry Level 5

Define \[F_n(x) = \prod_{k=1}^n (\sin^{2^k} x+\cos^{2^k} x)\]

\[\text{If}\ F_{2015}\left(\frac{\pi}{6}\right) = \frac{a^{2^{b-1}}-1}{c^{2^b-1}}\]

Then find the value of \(a+b+c\).

Clarification

  • The power of \(a\) in numerator is \(2^{b-1}\) whereas the power of \(c\) in denominator is \(2^b-1\)

  • \(a,\ b,\ c\) are positive integers.

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