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The given expression is:

\[
\frac{\cot x + \csc x}{\sin x + \tan x}
\]

You are asked to find which of the following expressions matches this identity from the provided options:

A. \( \sec x \cot x \)  
B. \( \csc x \cot x \)  
C. \( \cos x \sin x \)  
D. \( \tan^2 x - 1 \)

### Step-by-step Solution:

#### Step 1: Simplify the numerator and denominator separately

- The **numerator**: \( \cot x + \csc x \)

  - Recall that \( \cot x = \frac{\cos x}{\sin x} \) and \( \csc x = \frac{1}{\sin x} \), so:
  \[
  \cot x + \csc x = \frac{\cos x}{\sin x} + \frac{1}{\sin x} = \frac{\cos x + 1}{\sin x}
  \]

- The **denominator**: \( \sin x + \tan x \)

  - Recall that \( \tan x = \frac{\sin x}{\cos x} \), so:
  \[
  \sin x + \tan x = \sin x + \frac{\sin x}{\cos x} = \frac{\sin x \cos x + \sin x}{\cos x} = \frac{\sin x (\cos x + 1)}{\cos x}
  \]

#### Step 2: Substitute the simplified forms into the original expression

\[
\frac{\cot x + \csc x}{\sin x + \tan x} = \frac{\frac{\cos x + 1}{\sin x}}{\frac{\sin x (\cos x + 1)}{\cos x}} = \frac{\cos x + 1}{\sin x} \times \frac{\cos x}{\sin x (\cos x + 1)}
\]

#### Step 3: Cancel common terms

The \( \cos x + 1 \) cancels out from both the numerator and denominator, leaving:

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

#### Step 4: Recognize the final form

We know that \( \frac{\cos x}{\sin^2 x} = \cot x \csc x \).

Thus, the correct answer is **Option B**: \( \csc x \cot x \).

---

Would you like more details or have any questions? Here are some related questions to explore further:

1. What is the relationship between the trigonometric functions \( \cot x \) and \( \csc x \)?
2. How do you derive the identity \( \tan^2 x - 1 = \sec^2 x \)?
3. What are the steps for simplifying complex trigonometric expressions?
4. How does the Pythagorean identity relate to other trigonometric identities?
5. What is the significance of the angle sum and difference identities in simplifying expressions?

**Tip:** Always check for opportunities to cancel common terms when simplifying trigonometric expressions.

The expression cot x + csc x over sin x + tan 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?

Explore how to simplify the trigonometric expression cot x + csc x over sin x + tan x and match it with the correct trigonometric identity from multiple choices. This guide covers essential trigonometric identities and simplification techniques, suitable for high school students.

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