<Question>
Simplify
\[\frac{\sec x}{\sin x} - \frac{\sin x}{\cos x}.\]
</Question>

<think>
<Plan_1>
Okay, let's see. I need to simplify the expression sec x over sin x minus sin x over cos x. Hmm, first, I should recall what sec x is. Right, sec x is the reciprocal of cos x. So sec x is 1 over cos x. Let me write that down.

So, replacing sec x with 1/cos x, the first term becomes (1/cos x) divided by sin x. Dividing by sin x is the same as multiplying by 1/sin x, so that would be 1/(cos x * sin x). Let me check that: yes, (1/cos x)/sin x = 1/(cos x sin x).

The second term is sin x over cos x. That's straightforward. So the expression is now [1/(cos x sin x)] - [sin x / cos x].

Now, to combine these two terms, they need a common denominator. The first term already has the denominator cos x sin x, but the second term has denominator cos x. So I can multiply the numerator and denominator of the second term by sin x to get the same denominator.

<prompt_1.1> Determine the common denominator for the terms and adjust the second term accordingly. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Let me do that: sin x / cos x times sin x / sin x is (sin x * sin x)/(cos x * sin x) = sin²x / (cos x sin x).

So now the expression is [1 - sin²x]/(cos x sin x).
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, the numerator is 1 - sin²x. Oh, right! That's a Pythagorean identity.
1 - sin²x equals cos²x. So substituting that in, the numerator becomes cos²x. Therefore, the expression is now cos²x/(cos x sin x).

Simplifying the fraction, cos²x over cos x is cos x. So cos²x divided by cos x is cos x. So now we have cos x / sin x.
That's the definition of cotangent. So cos x / sin x is cot x.

Wait, let me check my steps again to make sure I didn't make a mistake.

<prompt_2.1> Review each step of the simplification process to verify its correctness. </prompt_2.1>

Is there another way to approach this? Maybe by combining the fractions first.

<prompt_2.2> Explore an alternative method to simplify the expression by combining the fractions first. </prompt_2.2>

Alternatively, could I have simplified the original expression differently? Let's see.

<prompt_2.3> Consider other potential simplification methods and confirm the result. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Starting with sec x / sin x - sin x / cos x. Sec x is 1/cos x, so first term is (1/cos x)/sin x = 1/(cos x sin x). Second term: sin x / cos x. To subtract them, find common denominator. Multiply second term by sin x/sin x: (sin x * sin x)/(cos x * sin x) = sin²x/(cos x sin x). Then subtract: [1 - sin²x]/(cos x sin x). Then 1 - sin²x is cos²x, so cos²x/(cos x sin x) simplifies to cos x / sin x, which is cot x.
</execution_2.1>

<execution_2.2>
Original expression: (sec x / sin x) - (sin x / cos x). Let's express everything in terms of sin x and cos x. Sec x is 1/cos x, so the first term is (1/cos x)/sin x = 1/(cos x sin x). The second term is sin x / cos x. So as before, combining them: 1/(cos x sin x) - sin x / cos x.

To combine these, the common denominator is cos x sin x. So for the second term, multiply numerator and denominator by sin x: (sin x * sin x)/(cos x sin x) = sin²x/(cos x sin x). Then subtract: [1 - sin²x]/(cos x sin x). Then 1 - sin²x is cos²x, so cos²x/(cos x sin x) = cos x / sin x = cot x.
</execution_2.2>

<execution_2.3>
Original expression: sec x / sin x - sin x / cos x.

Another approach: express everything in terms of sin x and cos x. We did that already. Sec x is 1/cos x, so first term is (1/cos x)/sin x = 1/(sin x cos x). Second term is sin x / cos x. Then, as before, subtract them and get [1 - sin²x]/(sin x cos x) = cos²x/(sin x cos x) = cos x / sin x = cot x.

Is there a possibility I missed a step or made an error in algebra? Let's verify with a specific value of x. Let's pick x = 45 degrees, where sin x = cos x = √2/2.

Compute original expression: sec x / sin x - sin x / cos x.

Sec 45 is 1/(√2/2) = 2/√2 = √2. So first term: √2 / (√2/2) = √2 * 2/√2 = 2. Second term: (√2/2) / (√2/2) = 1. So the expression is 2 - 1 = 1.

Cot x at 45 degrees is 1, since cot 45 = 1/tan 45 = 1/1 = 1. So that checks out.

Another test: x = 30 degrees. sin 30 = 1/2, cos 30 = √3/2. Then sec x = 1/(√3/2) = 2/√3. First term: (2/√3)/(1/2) = (2/√3)*(2/1) = 4/√3. Second term: (1/2)/(√3/2) = (1/2)*(2/√3) = 1/√3. So the expression is 4/√3 - 1/√3 = 3/√3 = √3. Cot 30 is 1/tan 30 = 1/(1/√3) = √3. So that also checks out.

Therefore, the simplification seems correct.
</execution_2.3>
</Execution_2>

<Final_answer>
\[\boxed{\cot x}\]
</Final_answer>
</think>