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Let's analyze the given expression and solve the identity step by step.

The expression is:

\[
\frac{\cos x - \sin^2 x}{\cos x} \cdot \csc x
\]

We need to find which one of the given options completes the identity. Here’s the breakdown:

1. **Simplify the expression**:
   
   \[
   \frac{\cos x - \sin^2 x}{\cos x}
   \]
   
   Separate the terms:

   \[
   = \frac{\cos x}{\cos x} - \frac{\sin^2 x}{\cos x}
   \]
   
   This simplifies to:
   
   \[
   = 1 - \frac{\sin^2 x}{\cos x}
   \]
   
   Now recall that:
   
   \[
   \frac{\sin^2 x}{\cos x} = \sin x \cdot \tan x
   \]
   
   Therefore:
   
   \[
   1 - \sin x \cdot \tan x
   \]

2. **Multiply by \(\csc x\)**:
   
   Now multiply the simplified expression by \(\csc x\):
   
   \[
   (1 - \sin x \cdot \tan x) \cdot \csc x
   \]
   
   Recall that \(\csc x = \frac{1}{\sin x}\). So:

   \[
   = \frac{1 - \sin x \cdot \tan x}{\sin x}
   \]
   
   Separate the terms:

   \[
   = \frac{1}{\sin x} - \frac{\sin x \cdot \tan x}{\sin x}
   \]

   Simplify each term:

   \[
   = \csc x - \tan x
   \]

3. **Compare with the options**:
   Now that we have simplified the left-hand side to \(\csc x - \tan x\), we can match it with the provided options. The correct answer is:

   \[
   \boxed{C. \csc x - \tan x}
   \]

Would you like further clarification on any step?

Here are five follow-up questions you might find useful:

1. How is \(\tan x\) related to \(\sin x\) and \(\cos x\)?
2. Can you derive the identity for \(\csc x\) in terms of sine?
3. What happens if you simplify \(\sin^2 x + \cos^2 x\)?
4. How does the distributive property help in simplifying trigonometric expressions?
5. Can you explain how multiplying by \(\csc x\) affects the original expression?

**Tip**: Always break down complex trigonometric expressions by separating terms and using fundamental identities (like \(\sin^2 x + \cos^2 x = 1\)) for easier manipulation!

The expression \( \frac{\cos x - \sin^2 x}{\cos x} \cdot \csc x \) is to be the left-hand side of an equation that is an identity. Which one of the following four expressions can be used as the right-hand side of the equation to complete the identity?

This problem focuses on simplifying a trigonometric expression and finding the correct identity. It involves key concepts like the Pythagorean identity, the relationship between sine, cosine, and tangent, and multiplying by the cosecant function. Suitable for high school students in Grades 10-12.

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