Alright, let's tackle this trigonometric expression simplification step by step. The problem is to simplify \(\frac{\sec x}{\sin x} - \frac{\sin x}{\cos x}\). Hmm, okay. Let me recall some trigonometric identities that might help here.

First, I know that \(\sec x\) is the reciprocal of \(\cos x\), so \(\sec x = \frac{1}{\cos x}\). Maybe rewriting \(\sec x\) in terms of \(\cos x\) will make things clearer. Let me try that.

Starting with the first term, \(\frac{\sec x}{\sin x}\), substituting \(\sec x\) gives \(\frac{\frac{1}{\cos x}}{\sin x}\). Dividing by \(\sin x\) is the same as multiplying by \(\frac{1}{\sin x}\), so that becomes \(\frac{1}{\cos x \cdot \sin x}\). Okay, so the first term simplifies to \(\frac{1}{\sin x \cos x}\).

The second term is \(\frac{\sin x}{\cos x}\), which is actually the definition of \(\tan x\). So that simplifies to \(\tan x\). Therefore, the entire expression becomes \(\frac{1}{\sin x \cos x} - \tan x\).

Hmm, maybe I can express both terms with a common denominator or find a way to combine them. Let me think. The first term is \(\frac{1}{\sin x \cos x}\), and the second term is \(\tan x = \frac{\sin x}{\cos x}\). To combine these, I might need a common denominator. Let's see.

The common denominator for \(\sin x \cos x\) and \(\cos x\) would be \(\sin x \cos x\). So, let's rewrite the second term with that denominator. Multiplying numerator and denominator by \(\sin x\), we get \(\frac{\sin x \cdot \sin x}{\cos x \cdot \sin x}\) which is \(\frac{\sin^2 x}{\sin x \cos x}\). Therefore, the expression becomes:

\[
\frac{1}{\sin x \cos x} - \frac{\sin^2 x}{\sin x \cos x} = \frac{1 - \sin^2 x}{\sin x \cos x}
\]

Ah, I remember that \(1 - \sin^2 x\) is equal to \(\cos^2 x\). So substituting that in, the numerator becomes \(\cos^2 x\). Therefore, the expression simplifies to:

\[
\frac{\cos^2 x}{\sin x \cos x}
\]

Simplifying further, the \(\cos x\) in the numerator and denominator cancels out, leaving:

\[
\frac{\cos x}{\sin x}
\]

And \(\frac{\cos x}{\sin x}\) is the definition of \(\cot x\). So, the simplified form of the original expression is \(\cot x\).

Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from the original expression:

\[
\frac{\sec x}{\sin x} - \frac{\sin x}{\cos x}
\]

Substituting \(\sec x = \frac{1}{\cos x}\):

\[
\frac{1}{\cos x \cdot \sin x} - \frac{\sin x}{\cos x}
\]

Finding a common denominator:

\[
\frac{1}{\sin x \cos x} - \frac{\sin^2 x}{\sin x \cos x} = \frac{1 - \sin^2 x}{\sin x \cos x}
\]

Since \(1 - \sin^2 x = \cos^2 x\), substitute:

\[
\frac{\cos^2 x}{\sin x \cos x} = \frac{\cos x}{\sin x} = \cot x
\]

Yes, that seems correct. Alternatively, maybe there's another way to approach this problem. Let me try another method to confirm.

Starting again with the original expression:

\[
\frac{\sec x}{\sin x} - \frac{\sin x}{\cos x}
\]

Express both terms with sine and cosine:

First term: \(\frac{\sec x}{\sin x} = \frac{\frac{1}{\cos x}}{\sin x} = \frac{1}{\sin x \cos x}\)

Second term: \(\frac{\sin x}{\cos x}\)

So the expression is:

\[
\frac{1}{\sin x \cos x} - \frac{\sin x}{\cos x}
\]

Factor out \(\frac{1}{\cos x}\):

\[
\frac{1}{\cos x} \left( \frac{1}{\sin x} - \sin x \right )
\]

Simplify inside the parentheses:

\[
\frac{1}{\sin x} - \sin x = \frac{1 - \sin^2 x}{\sin x} = \frac{\cos^2 x}{\sin x}
\]

Therefore, the entire expression becomes:

\[
\frac{1}{\cos x} \cdot \frac{\cos^2 x}{\sin x} = \frac{\cos x}{\sin x} = \cot x
\]

Same result. Okay, that confirms it. So regardless of the method, we arrive at \(\cot x\). Therefore, the simplified form is indeed the cotangent of x.

Just to ensure there's no alternative interpretation, let me check if there's a different identity or approach that might lead to a different simplification. For example, maybe using reciprocal identities or co-function identities. But in this case, since the result is a basic trigonometric function, cotangent, which is the reciprocal of tangent, and tangent is already involved in the original expression, I don't think there's a further simplification. Unless there's a way to express cotangent in terms of another function, but that's not necessary here. The problem just asks for simplification, and cotangent is a simpler form than the original expression.

Therefore, I can confidently say that the simplified form of the given expression is \(\cot x\).
Therefore, the simplified form of the given expression is \(\boxed{\cot x}\).
</think>