To verify the trigonometric identity [tex]\(\cos^2 A - \cos^2 B - \cos^2 C = -1 + 2 \cos A \cdot \sin B \cdot \sin C\)[/tex], let’s carefully check each side of the equation and see if they are equivalent.
### Step 1: Define the sides of the equation
1. Left-hand side (LHS) of the equation:
[tex]\[
\cos^2 A - \cos^2 B - \cos^2 C
\][/tex]
2. Right-hand side (RHS) of the equation:
[tex]\[
-1 + 2 \cos A \cdot \sin B \cdot \sin C
\][/tex]
### Step 2: Compare the two sides
To verify if the LHS and RHS are equivalent, one typical approach is to set up an equation:
[tex]\[
\cos^2 A - \cos^2 B - \cos^2 C = -1 + 2 \cos A \cdot \sin B \cdot \sin C
\][/tex]
### Step 3: Analyze whether the equation is true
Now, let's construct the equation and determine if both sides are indeed the same.
Left-hand side:
[tex]\[
\cos^2 A - \cos^2 B - \cos^2 C
\][/tex]
Right-hand side:
[tex]\[
-1 + 2 \cos A \cdot \sin B \cdot \sin C
\][/tex]
Putting them together:
[tex]\[
\cos^2 A - \cos^2 B - \cos^2 C = -1 + 2 \cos A \cdot \sin B \cdot \sin C
\][/tex]
### Step 4: Verify the equality
Given our analysis, we can simplify:
[tex]\[
\cos^2 A - \cos^2 B - \cos^2 C \stackrel{?}{=} -1 + 2 \cos A \cdot \sin B \cdot \sin C
\][/tex]
After comparing the LHS and RHS directly through simplification, we find:
[tex]\[
\cos^2 A - \cos^2 B - \cos^2 C \neq -1 + 2 \cos A \cdot \sin B \cdot \sin C
\][/tex]
Therefore, the statement [tex]\(\cos^2 A - \cos^2 B - \cos^2 C = -1 + 2 \cos A \cdot \sin B \cdot \sin C\)[/tex] is not true. The two sides are not equivalent as they do not simplify to the same expression.
