Let’s start by simplifying each part of this problem step-by-step.

1. Simplifying the First Expression:

$\frac{\sin(90^\circ - x)}{\sin(360^\circ - x)} \div \tan(x - 180^\circ)$

Step 1.1: Simplify $\sin(90^\circ - x)$

Using the identity $\sin(90^\circ - x) = \cos(x)$: $\sin(90^\circ - x) = \cos(x)$

Step 1.2: Simplify $\sin(360^\circ - x)$

Using the identity $\sin(360^\circ - x) = -\sin(x)$: $\sin(360^\circ - x) = -\sin(x)$

Thus, the first part simplifies as follows: $\frac{\sin(90^\circ - x)}{\sin(360^\circ - x)} = \frac{\cos(x)}{-\sin(x)} = -\cot(x)$

Step 1.3: Simplify $\tan(x - 180^\circ)$

Using the identity $\tan(x - 180^\circ) = \tan(x)$: $\tan(x - 180^\circ) = \tan(x)$

Now, we have: $-\cot(x) \div \tan(x)$

Since $\cot(x) = \frac{1}{\tan(x)}$, this division simplifies to: $-\cot(x) \div \tan(x) = -\cot(x) \cdot \frac{1}{\tan(x)} = -1$

2. Simplifying the Second Expression:

$\frac{\tan(205^\circ) \cdot \cos(315^\circ) \cdot \sin(135^\circ)}{\sin(210^\circ) \cdot \cos(150^\circ) \cdot \tan(25^\circ)}$

Step 2.1: Evaluate Each Trigonometric Function

Using known trigonometric values:

• $\tan(205^\circ) = \tan(180^\circ + 25^\circ) = \tan(25^\circ)$
• $\cos(315^\circ) = \cos(360^\circ - 45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$
• $\sin(135^\circ) = \sin(180^\circ - 45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2}$
• $\sin(210^\circ) = \sin(180^\circ + 30^\circ) = -\sin(30^\circ) = -\frac{1}{2}$
• $\cos(150^\circ) = \cos(180^\circ - 30^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}$
• $\tan(25^\circ) = \tan(25^\circ)$

Step 2.2: Substitute and Simplify

Substitute these values into the expression: $\frac{\tan(25^\circ) \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}}{-\frac{1}{2} \cdot -\frac{\sqrt{3}}{2} \cdot \tan(25^\circ)}$

Simplify each part: $= \frac{\tan(25^\circ) \cdot \frac{2}{4}}{\frac{\sqrt{3}}{4} \cdot \tan(25^\circ)}$

The $\tan(25^\circ)$ terms cancel, and we get: $= \frac{\frac{2}{4}}{\frac{\sqrt{3}}{4}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$

Final Answer:

The simplified expressions are:

1. $-1$
2. $\frac{2\sqrt{3}}{3}$

Would you like further details or clarification on any steps?

Here are some related questions to explore:

1. What are the co-function identities in trigonometry?
2. How do transformations affect trigonometric function values?
3. Can trigonometric functions of negative angles be simplified?
4. How does the ASTC rule (All Students Take Calculus) help with trigonometric simplifications?
5. Why does $\tan(x - 180^\circ) = \tan(x)$?

Tip: Remember that trigonometric identities often simplify expressions significantly, especially in calculations without calculators.