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

# Prove That: Sin a + Sin 3 a Cos a − Cos 3 a = Cot a - Mathematics

Sum

Prove that:

$\frac{\sin A + \sin 3A}{\cos A - \cos 3A} = \cot A$

#### Solution

Consider LHS:
$\frac{\sin A + \sin 3A}{\cos A - \cos 3A}$
$= \frac{2\sin \left( \frac{A + 3A}{2} \right) \cos \left( \frac{A - 3A}{2} \right)}{2\sin \left( \frac{A + 3A}{2} \right) \sin \left( \frac{3A - A}{2} \right)} \left\{ \because \sin A + \sin B = 2\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right), and \cos A - \cos B = 2\sin \left( \frac{A + B}{2} \right) cos \left( \frac{B - A}{2} \right) \right\}$
$= \frac{\sin 2A \cos \left( - A \right)}{\sin 2A \sin A}$
$= \frac{\sin 2A \cos A}{\sin 2A \sin A}$
$= \cot A$
= 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 7.1 | Page 18