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To prove the identity \(\cot x \cos x + \sin x = \csc x\), let's go step-by-step using appropriate rules from the provided options.

### 1. Simplify \(\cot x \cos x\) using the quotient rule:
\[
\cot x = \frac{\cos x}{\sin x}
\]
So,
\[
\cot x \cos x = \frac{\cos x}{\sin x} \cdot \cos x = \frac{\cos^2 x}{\sin x}
\]

### 2. Add \(\sin x\) to the expression:
\[
\frac{\cos^2 x}{\sin x} + \sin x
\]

### 3. Get a common denominator:
\[
\frac{\cos^2 x}{\sin x} + \frac{\sin^2 x}{\sin x} = \frac{\cos^2 x + \sin^2 x}{\sin x}
\]

### 4. Use the Pythagorean identity:
\[
\cos^2 x + \sin^2 x = 1
\]
So the expression simplifies to:
\[
\frac{1}{\sin x}
\]

### 5. Finally, use the reciprocal rule:
\[
\frac{1}{\sin x} = \csc x
\]

Thus, we have proved the identity:
\[
\cot x \cos x + \sin x = \csc x
\]

You can use the following rules step-by-step in your interface:

1. **Quotient Rule**: \(\cot x = \frac{\cos x}{\sin x}\)
2. **Algebra**: Multiplying and adding the terms.
3. **Sum and Difference**: Combine the terms under a common denominator.
4. **Pythagorean Identity**: \(\cos^2 x + \sin^2 x = 1\)
5. **Reciprocal**: \(\frac{1}{\sin x} = \csc x\)

Would you like more details on any of these steps, or have other questions?

### Related Questions
1. How do you simplify trigonometric expressions using identities?
2. What are common trigonometric identities to remember?
3. How can you prove an identity using algebraic manipulation?
4. What is the difference between quotient and reciprocal identities?
5. How do you choose the right identity to use in a proof?

### Tip
When proving trigonometric identities, always look for opportunities to use fundamental identities like Pythagorean, reciprocal, and quotient identities to simplify the expression.

Prove the identity cot x cos x + sin x = csc x.

Learn how to prove the trigonometric identity cot x cos x + sin x = csc x with a detailed step-by-step solution using the Pythagorean Identity, Reciprocal Identity, and Quotient Identity. This problem is suitable for high school students learning trigonometric identities.

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