Prove that: ^22 5 = ^2 3 = 5-1 8 - Vidyadip

Question

Prove that: $\sin^2\frac{2\pi}{5}=\sin^2\frac{\pi}{3}=\frac{\sqrt{5}-1}{8}$

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Answer

We have, $\sin^272^\circ-\sin^260^\circ,$ $=\sin^2(90^\circ-18^\circ)-\Big(\frac{\sqrt{3}}{2}\Big)^2$ $=\cos^218^\circ-\frac{3}{4}$ $\Bigg(\frac{\sqrt{10+2+\sqrt{5}}}{4}\Bigg)^2-\frac{3}{4}\Bigg[\because\cos18^\circ=\frac{\sqrt{10+2\sqrt{5}}}{4}\Bigg]$ $=\frac{10+2\sqrt{5}}{16}-\frac{3}{4}$ $=\frac{10+2\sqrt{5}-12}{16}$ $=\frac{2\sqrt{5}-2}{16}$ $=\frac{\sqrt{5}-1}{8}$

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