`cos^(2) A + cos^(2) B + cos^(2) C =`

Options

A. `1-2 sin A sin B sin C`
B. `1-2 cos A cos B cos C`
C. `1+sin A sin B sin C`
D. `1+ cos A cos B cos C `

Answer

To solve the equation \( \cos^2 A + \cos^2 B + \cos^2 C \), where \( A + B + C = 180^\circ \), we can use trigonometric identities and properties. Here’s a step-by-step solution:

### Step 1: Use the identity for cosine
We know that:
\[ \cos^2 A = 1 - \sin^2 A \]
\[ \cos^2 B = 1 - \sin^2 B \]
\[ \cos^2 C = 1 - \sin^2 C \]

### Step 2: Substitute the identities into the equation
Substituting these identities into the original equation gives us:
\[ \cos^2 A + \cos^2 B + \cos^2 C = (1 - \sin^2 A) + (1 - \sin^2 B) + (1 - \sin^2 C) \]

### Step 3: Simplify the expression
This simplifies to:
\[ = 3 - (\sin^2 A + \sin^2 B + \sin^2 C) \]

### Step 4: Use the identity for sine
Since \( A + B + C = 180^\circ \), we can use the identity:
\[ \sin^2 A + \sin^2 B + \sin^2 C = 2 \]
This is derived from the fact that \( \sin^2 A + \sin^2 B + \sin^2(180^\circ - (A + B)) = 2 \).

### Step 5: Substitute back into the equation
Now substituting \( \sin^2 A + \sin^2 B + \sin^2 C = 2 \) into our expression gives:
\[ \cos^2 A + \cos^2 B + \cos^2 C = 3 - 2 = 1 \]

### Final Answer
Thus, we conclude that:
\[ \cos^2 A + \cos^2 B + \cos^2 C = 1 \]

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