Question

In ΔABC, prove that $\frac{a}{bc} + \frac{cos\ A}{a}\ =\ \frac{b}{ca} + \frac{cos\ B}{b}\ =\ \frac{c}{ab} + \frac{cos\ C}{c}$.

Answer 1

Answer:

$\frac{a}{bc}+\frac{cosA}{a}=\frac{b}{ca}+\frac{cosB}{b}=\frac{c}{ab}+\frac{cosC}{c}$

Step-by-step explanation:

Formula used:

Projection formula:

In triangle ABC,

a = b cosC + c cosB

b = c cosB + b cosC

c = a cosB + b cosA

$\frac{a}{bc}+\frac{cosA}{a}$

$=\frac{[b\:cosC +c\:cosB}{bc}+\frac{cosA}{a}$

$=\frac{b\:cosC}{bc}+\frac{c\:cosB}{bc}+\frac{cosA}{a}$

$=\frac{cosC}{c}+\frac{cosB}{b}+\frac{cosA}{a}$

$=\frac{cosA}{a}+\frac{cosB}{b}+\frac{cosC}{c}$

similarly we can prove$\frac{b}{ca}+\frac{cosB}{b}=\frac{cosA}{a}+\frac{cosB}{b}+\frac{cosC}{c}$

$\frac{c}{ab}+\frac{cosC}{c}=\frac{cosA}{a}+\frac{cosB}{b}+\frac{cosC}{c}$

$\frac{a}{bc}+\frac{cosA}{a}=\frac{b}{ca}+\frac{cosB}{b}=\frac{c}{ab}+\frac{cosC}{c}$