Prove that: 38^ + 22^ = 82^ - Vidyadip

Question

Prove that: $\sin38^\circ+\sin22^\circ=\sin82^\circ$

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Answer

$\sin38^\circ+\sin22^\circ=\sin82^\circ$ $\text{LHS}=\sin38^\circ+\sin22^\circ$ $\because\ \sin\text{C}\sin\text{D}=2\sin\frac{\text{C+D}}{2}\cos\frac{\text{C-D}}{2}$ $\Rightarrow\ \sin38^\circ+\sin22^\circ=2\sin\frac{60^\circ}{2}\cos\frac{16^\circ}{2}$ $=\ 2\sin30^\circ\cos8^\circ$ $=\ 2\times\frac{1}{2}\cos8^\circ$ $=\ \cos(90-82)^\circ$ $=\ \sin82^\circ=\text{RHS}$ $[\because\ \cos\text{x}=\sin(90-\text{x})]$

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