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

Question

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

✓

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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