# In ∆Abc, Prove the Following: 4 ( B C Cos 2 a 2 + C a Cos 2 B 2 + a B Cos 2 C 2 ) = ( a + B + C ) 2 - Mathematics

In ∆ABC, prove the following:

$4\left( bc \cos^2 \frac{A}{2} + ca \cos^2 \frac{B}{2} + ab \cos^2 \frac{C}{2} \right) = \left( a + b + c \right)^2$

#### Solution

$\text{ LHS }$

$= 4\left( bc \cos^2 \frac{A}{2} + ca \cos^2 \frac{B}{2} + ab \cos^2 \frac{C}{2} \right)$

$= 4\left[ bc\left( \frac{1 + \cos A}{2} \right) + ca\left( \frac{1 + \cos B}{2} \right) + ab\left( \frac{1 + \cos C}{2} \right) \right]$

$= 2bc + 2bc\cos A + 2ca + 2ca\cos B + 2ab + 2ab\cos C$

$= 2\left( ab + bc + ca \right) + 2bc\left( \frac{b^2 + c^2 - a^2}{2bc} \right) + 2ca\left( \frac{c^2 + a^2 - b^2}{2ca} \right) + 2ab\left( \frac{a^2 + b^2 - c^2}{2ab} \right)$

$= 2\left( ab + bc + ac \right) + b^2 + c^2 - a^2 + c^2 + a^2 - b^2 + a^2 + b^2 - c^2$

$= a^2 + b^2 + c^2 + 2ab + 2bc + 2ac$

$= \left( a + b + c \right)^2 = RHS$

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 10.2 | Q 13 | Page 25