If $\cos A=\frac{5}{13}$ and $A$ is acute, find $\tan A$.

The correct option is $\frac{12}{5}$.

Step 1: Understand the given information

We are given that $\cos A=\frac{5}{13}$ and angle $A$ is acute (i.e., in the first quadrant, where all trigonometric functions are positive).

Step 2: Recall trigonometric definitions in a right-angled triangle

In a right-angled triangle with respect to an acute angle $A$:

$\cos A=\frac{\text{Adjacent Side}}{\text{Hypotenuse}}$

$\tan A=\frac{\text{Opposite Side}}{\text{Adjacent Side}}$

Step 3: Determine the sides of the triangle

Let the Adjacent Side be $5k$ and the Hypotenuse be $13k$, where $k>0$.

Using the Pythagoras Theorem:

${\left(\text{Opposite Side}\right)}^2+{\left(\text{Adjacent Side}\right)}^2={\left(\text{Hypotenuse}\right)}^2$

Substitute the given values into the formula:

${\left(\text{Opposite Side}\right)}^2+{\left(5k\right)}^2={\left(13k\right)}^2$

${\left(\text{Opposite Side}\right)}^2+25k^2=169k^2$

${\left(\text{Opposite Side}\right)}^2=169k^2-25k^2$

${\left(\text{Opposite Side}\right)}^2=144k^2$

Taking the square root on both sides:

$\text{Opposite Side}=12k$

Step 4: Calculate $\tan A$

Now, substitute the values of the Opposite Side and Adjacent Side into the ratio for $\tan A$:

$\tan A=\frac{12k}{5k}=\frac{12}{5}$

Thus, the value of $\tan A$ is $\frac{12}{5}$.