To simplify [tex]\(\frac{\sec x}{\cos x} - \frac{\tan x}{\cot x}\)[/tex], let's start by breaking down each term:
1. Simplifying [tex]\(\frac{\sec x}{\cos x}\)[/tex]:
- We know that [tex]\(\sec x = \frac{1}{\cos x}\)[/tex].
- Thus, [tex]\(\frac{\sec x}{\cos x} = \frac{\frac{1}{\cos x}}{\cos x} = \frac{1}{\cos x} \cdot \frac{1}{\cos x} = \frac{1}{\cos^2 x} = \cos(x)^{-2}\)[/tex].
2. Simplifying [tex]\(\frac{\tan x}{\cot x}\)[/tex]:
- We know that [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex] and [tex]\(\cot x = \frac{\cos x}{\sin x}\)[/tex].
- Thus, [tex]\(\frac{\tan x}{\cot x} = \frac{\frac{\sin x}{\cos x}}{\frac{\cos x}{\sin x}} = \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} = \frac{\sin^2 x}{\cos^2 x}\)[/tex].
3. Combining the two terms:
- From the simplified results, we have:
[tex]\[
\frac{\sec x}{\cos x} - \frac{\tan x}{\cot x} = \frac{1}{\cos^2 x} - \frac{\sin^2 x}{\cos^2 x}.
\][/tex]
- We can combine these fractions by having a common denominator:
[tex]\[
\frac{1}{\cos^2 x} - \frac{\sin^2 x}{\cos^2 x} = \frac{1 - \sin^2 x}{\cos^2 x}.
\][/tex]
4. Simplifying the final expression:
- Recall the Pythagorean identity: [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex].
- Thus, [tex]\(1 - \sin^2 x = \cos^2 x\)[/tex].
- Thus, the expression simplifies to:
[tex]\[
\frac{1 - \sin^2 x}{\cos^2 x} = \frac{\cos^2 x}{\cos^2 x} = 1.
\][/tex]
So, the simplified form of [tex]\(\frac{\sec x}{\cos x} - \frac{\tan x}{\cot x}\)[/tex] is [tex]\(\boxed{1}\)[/tex].
