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To simplify these expressions, let’s analyze each part separately and apply trigonometric identities where possible. Here is a breakdown for each term:

### 1. Simplify \( \frac{\sin(90^\circ - x)}{\sin(360^\circ - x)} \)
Using trigonometric identities:
- **For \( \sin(90^\circ - x) \)**: This is equal to \( \cos(x) \) since \( \sin(90^\circ - x) = \cos(x) \).
- **For \( \sin(360^\circ - x) \)**: This is equal to \( -\sin(x) \), as \( \sin(360^\circ - x) = -\sin(x) \).

Thus,
\[
\frac{\sin(90^\circ - x)}{\sin(360^\circ - x)} = \frac{\cos(x)}{-\sin(x)} = -\cot(x).
\]

### 2. Simplify \( \tan(x - 180^\circ) \)
Using the identity for tangent:
- **For \( \tan(x - 180^\circ) \)**: This is equal to \( \tan(x) \) because \( \tan(x - 180^\circ) = \tan(x) \).

### Combined Expression
Putting it all together, we get:
\[
\frac{\sin(90^\circ - x)}{\sin(360^\circ - x)} + \tan(x - 180^\circ) = -\cot(x) + \tan(x).
\]

This is the simplified form of the expression. 

Would you like additional steps or explanations on specific parts?

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Here are some related questions for practice:
1. How would you simplify \( \cos(90^\circ - x) \) using trigonometric identities?
2. What is the result of \( \sin(180^\circ - x) \)?
3. How does the identity \( \tan(180^\circ + x) \) relate to \( \tan(x) \)?
4. Can you simplify \( \cos(360^\circ - x) \) in a similar way?
5. What is the general rule for simplifying expressions with angles like \( 90^\circ - x \) and \( 360^\circ - x \)?

**Tip:** Remember that transformations of angles around 90°, 180°, and 360° often have equivalent values based on symmetry properties of the unit circle.

Simplify the following expression without the use of a calculator: \( \frac{\sin(90^\circ - x)}{\sin(360^\circ - x)} + \tan(x - 180^\circ) \)

Learn how to simplify the trigonometric expression \( \frac{\sin(90^\circ - x)}{\sin(360^\circ - x)} + \tan(x - 180^\circ) \) using trigonometric identities and angle transformations. This step-by-step solution covers important trigonometric concepts and identities, suitable for high school students.

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