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?

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.