Prove that: 5 18 - 4 9 = 3 9 - Vidyadip

Question

Prove that: $\sin\frac{5\pi}{18}-\cos\frac{4\pi}{9}=\sqrt3\sin\frac{\pi}{9}$

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Answer

$\sin\frac{5\pi}{18}-\cos\frac{4\pi}{9}=\sqrt3\sin\frac{\pi}{9}$ $\text{LHS}=\sin\frac{5\pi}{18}-\cos\frac{4\pi}{9}$ $=\ \sin50^\circ-\cos80^\circ$ $=\ \sin50^\circ-\sin10^\circ$ $=\ 2\sin\Big(\frac{50^\circ-10^\circ}{2}\Big)\cos\Big(\frac{50^\circ+10^\circ}{2}\Big)$ $=\ 2\sin20^\circ\cos30^\circ$ $=\ 2\sin20^\circ\times\frac{\sqrt3}{2}$ $=\ \sqrt3\sin\frac{\pi}{9}=\text{RHS}$

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