In a ∆ABC, if

Using cosine formula

$\cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c}, \cos B=\frac{a^{2}+c^{2}-b^{2}}{2 a c}$ and $\cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}$

L. H. S.

$=\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}$

$=\frac{b^{2}+c^{2}-a^{2}}{2 b c} \times \frac{1}{a}+\frac{a^{2}+c^{2}-b^{2}}{2 a c} \times \frac{1}{b}+\frac{a^{2}+b^{2}-c^{2}}{2 a b} \times \frac{1}{c}$

$=\frac{b^{2}+c^{2}-a^{2}+a^{2}+c^{2}-b^{2}+a^{2}+b^{2}-c^{2}}{2 a b c}$

$=\frac{a^{2}+b^{2}+c^{2}}{2 a b c}=k\left(a^{2}+b^{2}+c^{2}\right)$

$=$ R. H.S (given)

$\Rightarrow k=\frac{1}{2 a b c}$