To simplify the expression [tex]\(\frac{\sec x}{\tan x}\)[/tex], let's break it down step-by-step using trigonometric identities.
1. First, recall the definitions of the trigonometric functions involved:
- [tex]\(\sec x = \frac{1}{\cos x}\)[/tex]
- [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex]
2. Substitute these identities into the given expression:
[tex]\[
\frac{\sec x}{\tan x} = \frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}}
\][/tex]
3. Simplify the expression by dividing the numerators and the denominators:
[tex]\[
\frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}} = \frac{1}{\cos x} \times \frac{\cos x}{\sin x}
\][/tex]
4. The [tex]\(\cos x\)[/tex] terms cancel each other out:
[tex]\[
\frac{1}{\sin x}
\][/tex]
5. Recall the definition of the cosecant function:
- [tex]\(\csc x = \frac{1}{\sin x}\)[/tex]
So, the simplified form of the expression [tex]\(\frac{\sec x}{\tan x}\)[/tex] is [tex]\(\csc x\)[/tex].
Therefore, the correct choice is:
b. [tex]\(\csc x\)[/tex]
