Given that angles A and B are complementary and $\cos A=\frac{5}{13}$, what is $\tan B$?

The correct answer is Option 2: $\frac{5}{12}$.

Step 1: Understand complementary angles
Two angles, $A$ and $B$, are complementary if their sum is $90°$. Therefore, we can express $B$ in terms of $A$ as:

$B=90°-A$

Step 2: Relate trigonometric functions of complementary angles
Using the co-function and complementary angle identities for trigonometric functions:

$\sin B=\sin (90°-A)=\cos A$

$\cos B=\cos (90°-A)=\sin A$

We are given that:

$\cos A=\frac{5}{13}$

Hence, $\sin B=\frac{5}{13}$.

Step 3: Calculate $\sin A$ or use a right-angled triangle
Using the Pythagorean identity ${\sin }^{2}A+{\cos }^{2}A=1$:

${\sin }^{2}A=1-{\left(\frac{5}{13}\right)}^2=1-\frac{25}{169}=\frac{144}{169}$

Taking the positive square root for an acute angle $A$:

$\sin A=\frac{12}{13}$

Thus, $\cos B=\sin A=\frac{12}{13}$.

Step 4: Find $\tan B$
The tangent of an angle is defined as the ratio of its sine to its cosine:

$\tan B=\frac{\sin B}{\cos B}=\frac{\cos A}{\sin A}=\cot A$

Substituting the values of $\sin B$ and $\cos B$:

$\tan B=\frac{\frac{5}{13}}{\frac{12}{13}}=\frac{5}{12}$

Therefore, the value of $\tan B$ is $\frac{5}{12}$.