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

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

Short Note
Sum

Prove that:

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

#### Solution

Consider LHS:
$\frac{\cos A + cos B}{\cos B - \cos A}$
$= \frac{2\cos \left( \frac{A - B}{2} \right) \cos \left( \frac{A + B}{2} \right)}{2\sin \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)} \left[ \because \cos A + \cos B = 2\cos \left( \frac{A - B}{2} \right) \cos \left( \frac{A + B}{2} \right) and \cos A - \cos B = 2\sin \left( \frac{A + B}{2} \right) cos \left( \frac{B - A}{2} \right) \right]$
$= \frac{\cos \left( \frac{A - B}{2} \right) \cos \left( \frac{A + B}{2} \right)}{\sin \left( \frac{A + B}{2} \right) sin \left( \frac{A - B}{2} \right)}$
$= \cot\left( \frac{A + B}{2} \right)\cot\left( \frac{A - B}{2} \right)$
=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 7.5 | Page 18