Prove that: ^2 18 + ^2 9 + ^27 18 + ^24 9 =2 - Vidyadip

Question

Prove that: $\sin^2\frac{\pi}{18}+\sin^2\frac{\pi}{9}+\sin^2\frac{7\pi}{18}+\sin^2\frac{4\pi}{9}=2$

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Answer

$\text{L.H.S}=\sin^2\frac{\pi}{18}+\sin^2\frac{\pi}{9}+\sin^2\frac{7\pi}{18}+\sin^2\frac{4\pi}{9}$ $=\sin^2\frac{\pi}{18}+\sin^2\frac{4\pi}{9}+\sin^2\frac{\pi}{9}+\sin^2\frac{7\pi}{18}$ $=\sin^2\Big(\frac{\pi}{2}-\frac{4\pi}{9}\Big)+\sin^2\frac{4\pi}{9}+\sin^2\frac{\pi}{9}+\sin^2\Big(\frac{\pi}{2}-\frac{\pi}{9}\Big)$ $\big(\because\frac{\pi}{18}=\frac{\pi}{2}-\frac{4\pi}{9}\text{ and }\frac{7\pi}{18}=\frac{\pi}{2}-\frac{\pi}{9}\big)$ $=\cos^2\frac{4\pi}{9}+\sin^2\frac{4\pi}{9}+\sin^2\frac{\pi}{9}+\cos^2\frac{\pi}{9}$ $=1+1$ $(\because\sin^2\theta+\cos^2\theta=1)$ $= 2$ $= \text{R.H.S}$ $\text{proved}$

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