tan(-225°)tan(180°-θ)cos(360°-θ)cos240°

To solve the expression \( \tan(-225^\circ) \tan(180^\circ - \theta) \cos(360^\circ - \theta) \cos 240^\circ \), let's evaluate each trigonometric function step-by-step:

### Step 1: Simplify Each Trigonometric Function

1. **Evaluate \( \tan(-225^\circ) \):**

   The tangent function has a period of \(180^\circ\), which means \( \tan(\theta) = \tan(\theta + 180^\circ) \). 

   - \( \tan(-225^\circ) = \tan(-225^\circ + 180^\circ) = \tan(-45^\circ) \).
   - \( \tan(-45^\circ) = -\tan(45^\circ) = -1 \).

   Therefore, \( \tan(-225^\circ) = -1 \).

2. **Evaluate \( \tan(180^\circ - \theta) \):**

   The tangent function is negative in the second quadrant (from \(90^\circ\) to \(180^\circ\)):

   - \( \tan(180^\circ - \theta) = -\tan(\theta) \).

3. **Evaluate \( \cos(360^\circ - \theta) \):**

   The cosine function has a period of \(360^\circ\), and it is positive in the fourth quadrant:

   - \( \cos(360^\circ - \theta) = \cos(\theta) \).

4. **Evaluate \( \cos 240^\circ \):**

   \(240^\circ\) is in the third quadrant, where cosine is negative:

   - \( \cos 240^\circ = \cos(180^\circ + 60^\circ) = -\cos(60^\circ) \).
   - \( \cos(60^\circ) = \frac{1}{2} \), so \( \cos 240^\circ = -\frac{1}{2} \).

### Step 2: Combine the Results

Now substitute these values back into the expression:

\[
\tan(-225^\circ) \tan(180^\circ - \theta) \cos(360^\circ - \theta) \cos 240^\circ = (-1) \cdot (-\tan(\theta)) \cdot \cos(\theta) \cdot \left(-\frac{1}{2}\right)
\]

Simplify this expression:

\[
(-1)(-\tan(\theta))(\cos(\theta))\left(-\frac{1}{2}\right) = -\frac{1}{2} \tan(\theta) \cos(\theta)
\]

Thus, the simplified expression is:

\[
\frac{1}{2} \tan(\theta) \cos(\theta)
\]

### Final Answer
\[
\frac{1}{2} \tan(\theta) \cos(\theta)
\]

Would you like more details on any step, or do you have any other questions?

### Follow-up Questions

1. What are the key properties of the tangent and cosine functions?
2. How can trigonometric identities simplify expressions involving sine and cosine?
3. Why is the tangent function negative in the second quadrant?
4. What are the general forms of trigonometric functions for any angle?
5. How do trigonometric functions behave in different quadrants?

**Tip:** Remember that recognizing which quadrant an angle falls into helps in determining the sign of trigonometric functions.

Learn how to solve the trigonometric expression tan(-225°) tan(180°-θ) cos(360°-θ) cos 240° step-by-step. This problem involves evaluating tangent and cosine functions at specific angles and understanding their properties in trigonometry. Suitable for advanced high school students and anyone interested in trigonometric identities.

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