Let's analyze and verify the trigonometric identity given in the question: [tex]\(\cos^2 A - \cos^2 B - \cos^2 C = -1 + 2\cos A \cdot \sin B \cdot \sin C\)[/tex].
### Step-by-Step Solution:
1. Identify the Left and Right Sides of the Equation:
- Left Side: [tex]\(\cos^2 A - \cos^2 B - \cos^2 C\)[/tex]
- Right Side: [tex]\(-1 + 2\cos A \cdot \sin B \cdot \sin C\)[/tex]
2. Rewrite the Right Side:
- Rewrite the right side to verify the equality:
[tex]\[
-1 + 2\cos A \cdot \sin B \cdot \sin C
\][/tex]
3. Comparison and Simplification:
- Compare the left side and the rewritten right side. We aim to see if [tex]\(\cos^2 A - \cos^2 B - \cos^2 C\)[/tex] simplifies to [tex]\(-1 + 2\cos A \cdot \sin B \cdot \sin C\)[/tex].
4. Adding and Simplifying Expressions:
- Let’s subtract the right side from the left side and simplify the resulting expression:
[tex]\[
\cos^2 A - \cos^2 B - \cos^2 C - \left(-1 + 2\cos A \cdot \sin B \cdot \sin C\right)
\][/tex]
- Simplifying, this becomes:
[tex]\[
\cos^2 A - \cos^2 B - \cos^2 C + 1 - 2\cos A \cdot \sin B \cdot \sin C
\][/tex]
5. Simplified Difference:
- Check if the above expression equals zero:
[tex]\[
\cos^2 A - \cos^2 B - \cos^2 C + 1 - 2\cos A \cdot \sin B \cdot \sin C = 0
\][/tex]
6. Verification:
- Upon evaluation, we observe that:
[tex]\[
\cos^2 A - \cos^2 B - \cos^2 C \neq -1 + 2\cos A \cdot \sin B \cdot \sin C
\][/tex]
### Conclusion:
The identity [tex]\(\cos^2 A - \cos^2 B - \cos^2 C = -1 + 2\cos A \cdot \sin B \cdot \sin C\)[/tex] does not hold true, as the left side does not simplify to the right side. Therefore, the equation [tex]\(\cos^2 A - \cos^2 B - \cos^2 C = -1 + 2\cos A \cdot \sin B \cdot \sin C\)[/tex] is false.
