A complete ripoff

Geometry Level 5

\[\dfrac{\sin^6{0^{\circ}}+\sin^6{10^{\circ}}+\sin^6{20^{\circ}}+\sin^6{30^{\circ}}+\ldots+\sin^6{180^{\circ}}}{\cos^6{0^{\circ}}+\cos^6{10^{\circ}}+\cos^6{20^{\circ}}+\cos^6{30^{\circ}}+\ldots+\cos^6{180^{\circ}}} \\ \\ \]

If the value of the above expression is equals to \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers, find the value of \(b-a\).

Bonus: For the general expression below, can you find a general formula for the answer, \(b_{n} - a_{n} \) for all positive integers \(n\)?

\[\dfrac{\sin^{2n}{0^{\circ}}+\sin^{2n}{10^{\circ}}+\sin^{2n}{20^{\circ}}+\sin^{2n}{30^{\circ}}+\ldots+\sin^{2n}{180^{\circ}}}{\cos^{2n}{0^{\circ}}+\cos^{2n}{10^{\circ}}+\cos^{2n}{20^{\circ}}+\cos^{2n}{30^{\circ}}+\ldots+\cos^{2n}{180^{\circ}}}\]

Inspiration.

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