Prove the identity.
cos(x − 𝜋) = −cos(x)

To prove the identity \( \cos(x - \pi) = -\cos(x) \), we'll use the cosine subtraction formula:

\[
\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)
\]

### Step-by-Step Proof:

1. **Apply the Cosine Subtraction Formula**:

   Let \( A = x \) and \( B = \pi \). Then,

   \[
   \cos(x - \pi) = \cos(x)\cos(\pi) + \sin(x)\sin(\pi)
   \]

2. **Substitute the Values of \( \cos(\pi) \) and \( \sin(\pi) \)**:

   We know that:
   \[
   \cos(\pi) = -1 \quad \text{and} \quad \sin(\pi) = 0
   \]
   Substituting these into the equation gives:
   \[
   \cos(x - \pi) = \cos(x)(-1) + \sin(x)(0)
   \]

3. **Simplify the Expression**:

   Simplifying the equation:
   \[
   \cos(x - \pi) = -\cos(x) + 0
   \]

   Therefore:
   \[
   \cos(x - \pi) = -\cos(x)
   \]

This completes the proof. The identity \( \cos(x - \pi) = -\cos(x) \) is indeed true.

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

Here are five related questions:
1. How does the identity \( \cos(x + \pi) \) relate to this proof?
2. What is the value of \( \cos(\pi) \) and why?
3. How can you derive the sine identity \( \sin(x - \pi) = -\sin(x) \)?
4. What is the significance of the angle \( \pi \) in trigonometric identities?
5. Can this identity be visualized on the unit circle?

**Tip:** Understanding the unit circle is crucial for visualizing and proving trigonometric identities.

Learn how to prove the identity \( \cos(x - \pi) = -\cos(x) \) using the cosine subtraction formula. This detailed step-by-step solution covers trigonometric concepts and formulas relevant to advanced high school students.

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