Using the identity

MATHEMATICS · Subjective · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Using the identity sin⁡4x=38−12cos⁡2x+18cos⁡4x\sin^4x=\frac{3}{8}-\frac{1}{2}\cos2x+\frac{1}{8}\cos4x, or otherwise, if sin⁡4(π7)+sin⁡4(3π7)+sin⁡4(5π7)=ab\sin^4\left(\frac{\pi}{7}\right)+\sin^4\left(\frac{3\pi}{7}\right)+\sin^4\left(\frac{5\pi}{7}\right)=\frac{a}{b}, where aa and bb are coprime, find (a−b)(a-b).

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