The ratio is given as, $a:b:c=3:5:7$.

Consider$a=3x, b=5x, c=7x$.

Now,

$\cos A+\cos B=\frac{b^2+c^2-a^2}{2bc}+\frac{a^2+c^2-b^2}{2ac}$

$=\frac{(5x{)}^2+(7x{)}^2-(3x{)}^2}{2\left(5x\right)\left(7x\right)}+\frac{(3x{)}^2+(7x{)}^2-(5x{)}^2}{2\left(3x\right)\left(7x\right)}$

$=\frac{65x^2}{70x^2}+\frac{33x^2}{42x^2}$

$=\frac{65}{70}+\frac{33}{42}$

$=\frac{12}{7}$