# In Triangle Abc, Prove the Following: C a − B = Tan ( a 2 ) + Tan ( B 2 ) Tan ( a 2 ) − Tan ( B 2 ) - Mathematics

In triangle ABC, prove the following:

$\frac{c}{a - b} = \frac{\tan\left( \frac{A}{2} \right) + \tan \left( \frac{B}{2} \right)}{\tan \left( \frac{A}{2} \right) - \tan \left( \frac{B}{2} \right)}$

#### Solution

Let

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k$                          ...(1)

We need to prove:

$\frac{c}{a - b} = \frac{\tan\left( \frac{A}{2} \right) + \tan \left( \frac{B}{2} \right)}{\tan \left( \frac{A}{2} \right) - \tan \left( \frac{B}{2} \right)}$

Consider

$LHS = \frac{c}{a - b}$
$= \frac{k\sin C}{k\left( \sin A - \sin B \right)} \left( \text{ using } \left( 1 \right) \right)$
$= \frac{2\sin\frac{C}{2}\cos\frac{C}{2}}{2\sin\left( \frac{A - B}{2} \right)\cos\left( \frac{A + B}{2} \right)}$
$= \frac{\sin\left( \frac{\pi - \left( A + B \right)}{2} \right)\cos\frac{C}{2}}{\sin\left( \frac{A - B}{2} \right)\cos\left( \frac{A + B}{2} \right)} \left( \because A + B + C = \pi \right)$
$= \frac{\cos\frac{C}{2}\cos\left( \frac{A + B}{2} \right)}{\sin\left( \frac{A - B}{2} \right)\cos\left( \frac{A + B}{2} \right)}$
$= \frac{\cos\frac{C}{2}}{\sin\left( \frac{A - B}{2} \right)} . . . \left( 2 \right)$

$RHS = \frac{\tan\frac{A}{2} + \tan\frac{B}{2}}{\tan\frac{A}{2} - \tan\frac{B}{2}}$
$= \frac{\frac{\sin\frac{A}{2}}{\cos\frac{A}{2}} + \frac{\sin\frac{B}{2}}{\cos\frac{B}{2}}}{\frac{\sin\frac{A}{2}}{\cos\frac{A}{2}} - \frac{\sin\frac{B}{2}}{\cos\frac{B}{2}}}$
$= \frac{\sin\frac{A}{2}\cos\frac{B}{2} + \sin\frac{B}{2}\cos\frac{A}{2}}{\sin\frac{A}{2}\cos\frac{B}{2} - \sin\frac{B}{2}\cos\frac{A}{2}}$
$= \frac{\sin\left( \frac{A + B}{2} \right)}{\sin\left( \frac{A - B}{2} \right)}$
$= \frac{\sin\left( \frac{\pi - C}{2} \right)}{\sin\left( \frac{A - B}{2} \right)}$
$= \frac{\cos\frac{C}{2}}{\sin\left( \frac{A - B}{2} \right)}$
$= LHS \left( \text{ from } \left( 2 \right) \right)$
$\text{ Hence proved } .$

Concept: Sine and Cosine Formulae and Their Applications
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 10 Sine and cosine formulae and their applications
Exercise 10.1 | Q 6 | Page 13