\tan \theta =\frac{12}{5},\text{what is the value of}\sec \theta ?

The correct option is 13/5.

We are given the value of trigonometric function:

$\tan \theta =\frac{12}{5}$

We need to find the value of $\sec \theta$.

Step 1: Use the Pythagorean Trigonometric Identity
Recall the fundamental trigonometric identity relating $\tan \theta$ and $\sec \theta$:

${\sec }^2\theta =1+{\tan }^2\theta$

Step 2: Substitute the Given Value
Substitute $\tan \theta =\frac{12}{5}$ into the identity:

${\sec }^2\theta =1+{\left(\frac{12}{5}\right)}^2$

${\sec }^2\theta =1+\frac{144}{25}$

Step 3: Simplify the Expression
Take a common denominator to add the numbers:

${\sec }^2\theta =\frac{25+144}{25}$

${\sec }^2\theta =\frac{169}{25}$

Step 4: Take the Square Root
Taking the square root on both sides:

$\sec \theta =\sqrt{\frac{169}{25}}=\frac{13}{5}$

Thus, the value of $\sec \theta$ is 13/5.