Prove that: - + = A-B 2 - Vidyadip

Question

Prove that: $\frac{\sin\text{A}-\sin\text{B}}{\cos\text{A}+\cos\text{B}}=\tan\frac{\text{A}-\text{B}}{2}$

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Answer

We have, $\text{LHS}=\frac{\sin\text{A}-\sin\text{B}}{\cos\text{A}+\cos\text{B}}$ $=\ \frac{2\cos\Big(\frac{\text{A}+\text{B}}{2}\Big)\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)}{2\cos\Big(\frac{\text{A}+\text{B}}{2}\Big)\cos\Big(\frac{\text{A}-\text{B}}{2}\Big)}$ $=\ \frac{\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)}{\cos\Big(\frac{\text{A}-\text{B}}{2}\Big)}$ $= \tan\Big(\frac{\text{A}-\text{B}}{2}\Big)$ $=\ \text{RHS}$ $\therefore\ \frac{\sin\text{A}-\sin\text{B}}{\cos\text{A}+\cos\text{B}}=\tan\Big(\frac{\text{A}-\text{B}}{2}\Big).$ Hence proved.

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