To solve the given trigonometric equation:
[tex]\[ \sin^2 A - \sin^2 B + \sin^2 C = 2 \sin A \cdot \cos B \cdot \sin C \][/tex]
we need to manipulate it using trigonometric identities to see if we can verify or simplify the expression.
### Step 1: Use the Pythagorean Identity
The Pythagorean identity for sine and cosine is:
[tex]\[ \sin^2(x) = 1 - \cos^2(x) \][/tex]
### Step 2: Substitute the Identity into the Equation
Applying the above identity to the terms [tex]\(\sin^2 A\)[/tex], [tex]\(\sin^2 B\)[/tex], and [tex]\(\sin^2 C\)[/tex]:
[tex]\[ \sin^2 A = 1 - \cos^2 A \][/tex]
[tex]\[ \sin^2 B = 1 - \cos^2 B \][/tex]
[tex]\[ \sin^2 C = 1 - \cos^2 C \][/tex]
Substitute these into the original equation:
[tex]\[ (1 - \cos^2 A) - (1 - \cos^2 B) + (1 - \cos^2 C) = 2 \sin A \cdot \cos B \cdot \sin C \][/tex]
### Step 3: Simplify the Left-Hand Side
Now simplify the left side of the equation:
[tex]\[ (1 - \cos^2 A) - (1 - \cos^2 B) + (1 - \cos^2 C) \][/tex]
Combine like terms:
[tex]\[ 1 - \cos^2 A - 1 + \cos^2 B + 1 - \cos^2 C \][/tex]
Simplify further:
[tex]\[ \cos^2 B - \cos^2 A + 1 - \cos^2 C \][/tex]
### Step 4: Consider the Right-Hand Side
The right side of the original equation remains the same:
[tex]\[ 2 \sin A \cdot \cos B \cdot \sin C \][/tex]
### Final Simplified Form
Rewriting the entire equation with the simplified left-hand side:
[tex]\[ \cos^2 B - \cos^2 A + 1 - \cos^2 C = 2 \sin A \cdot \cos B \cdot \sin C \][/tex]
This matches our derived equation, verifying the transformation and simplification. Therefore, the simplified form of the original equation is:
[tex]\[ \cos^2(B) - \cos^2(A) + 1 - \cos^2(C) = 2 \sin A \cdot \cos B \cdot \sin C \][/tex]
Thus, the transformed equation we have is:
[tex]\[ \cos^2(B) - \cos^2(A) + 1 - \cos^2(C) - 2 \sin A \cdot \cos B \cdot \sin C \][/tex]
This detailed manipulation shows the connection through trigonometric identities and verifies the relationship given in the problem.
