Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Prove That: Sin a + Sin B Sin a − Sin B = Tan ( a + B 2 ) Cot ( a − B 2 ) - Mathematics

Sum

Prove that:

$\frac{\sin A + \sin B}{\sin A - \sin B} = \tan \left( \frac{A + B}{2} \right) \cot \left( \frac{A - B}{2} \right)$

#### Solution

Consider LHS:
$\frac{\sin A + \sin B}{\sin A - \sin B}$
$= \frac{2\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)}{2\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)} \left\{ \because \sin A + \sin B = 2\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right), and \sin A - \sin B = 2\sin \left( \frac{A - B}{2} \right) \cos\left( \frac{A + B}{2} \right) \right\}$
$= \frac{\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)}{\sin \left( \frac{A - B}{2} \right) \cos \left( \frac{A + B}{2} \right)}$
$= \tan \left( \frac{A + B}{2} \right) cot \left( \frac{A - B}{2} \right)$
= RHS
Hence, LHS = RHS.

Concept: Transformation Formulae
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 8 Transformation formulae
Exercise 8.2 | Q 7.4 | Page 18