Problem 305

$(I)$ To prove the statement, we start with the given equation and apply the Law of Sines:

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

Based on the addition formula for sine, $\sin(A+B)=\sin C$, we can express this relationship as

$$\frac{\cos A \sin B + \cos B \sin A}{\sin A \sin B} = \frac{\sin(A+B)}{\sin A \sin B} = 1$$

Thus, by multiplying both sides by $\sin A \sin B$, we get

$$\boxed{\sin A \sin B = \sin C}$$

$(II)$ Given that $b^2 + c^2 - a^2 = \frac{6}{5}bc$, we can use the Law of Cosines to find $\cos A$:

$$a^2 = b^2 + c^2 - 2bc \cos A$$

Comparing the given equation with the one from the Law of Cosines, we can deduce that $2bc \cos A = \frac{4}{5}bc$. Hence,

$$\cos A = \frac{4}{5}bc \div 2bc = \frac{2}{5}$$

Using the Pythagorean identity, $\sin^2 A + \cos^2 A = 1$, we find:

$$\sin A = \sqrt{1 - \cos^2 A} = \sqrt{1 - \left(\frac{2}{5}\right)^2} = \sqrt{1 - \frac{4}{25}} = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}$$

Now, we return to our original equation and substitute $\cos A$ and $\sin A$:

$$\frac{\cos A}{\sin A} + \frac{\cos B}{\sin B} = 1 \implies \frac{\frac{2}{5}}{\frac{\sqrt{21}}{5}} + \frac{\cos B}{\sin B} = 1 \implies \frac{2}{\sqrt{21}} + \frac{\cos B}{\sin B} = 1$$

This implies:

$$\frac{\cos B}{\sin B} = 1 - \frac{2}{\sqrt{21}}$$

To find $\tan B$, which is $\frac{\sin B}{\cos B}$, we take the reciprocal of the above expression:

$$\tan B = \left(1 - \frac{2}{\sqrt{21}}\right)^{-1} = \boxed{\left(\frac{\sqrt{21}}{\sqrt{21} - 2}\right)}$$