Prove that: 50^ - 70^ + 10^ =0 - Vidyadip

Question

Prove that: $\sin50^\circ-\sin70^\circ+\sin10^\circ=0$

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Answer

$\text{LHS}=\sin50^\circ-\sin70^\circ+\sin10^\circ$ $(\sin50^\circ-\sin70^\circ)+\sin10^\circ$ $=\ \Big(2\sin\Big(\frac{50^\circ-70^\circ}{2}\Big)\cos\Big(\frac{50^\circ+70^\circ}{2}\Big)\Big)+\sin10^\circ$ $\Big[\because\ \sin\text{c}-\sin\text{D}=2\sin\Big(\frac{\text{C}-\text{D}}{2}\Big)\cos\Big(\frac{\text{C+D}}{2}\Big)\Big]$ $=\ 2\sin(-10^\circ)\cos60^\circ+\sin10^\circ$ $=\ -2\sin10^\circ\times\frac{1}{2}+\sin10^\circ$ $\Big[\because\ \cos60^\circ=\frac{1}{2}\Big]$ $=\ 0$ $=\ \text{RHS}$

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