Question

In a triangle \(ABC\), if

\[ (a - b)^2 \cos^2 \frac{C}{2} + (a + b)^2 \sin^2 \frac{C}{2} = a^2 + b^2, \]

then \( \cos A \) is:

• A. \(\cos B\)
• B. \(\sin C\)
• C. \(\sin B\)
• D. \(\cos C\)

Hint:

Use the cosine rule and known identities to transform trigonometric expressions, especially when dealing with complex geometric relationships.

Answer 1

The Correct Option is C

Step 1: Analyze the given equation.
Expand and simplify the given equation using trigonometric identities: \[ (a-b)^2 \cos^2 \frac{C}{2} + (a+b)^2 \sin^2 \frac{C}{2} = a^2 + b^2. \] Applying the half-angle formulas and simplifying can lead to insights about the relationship between sides \(a, b,\) and angles \(A, B, C\). Step 2: Apply cosine rule and simplify.
Using the cosine rule in triangle geometry and comparing it with the given equation could simplify to: \[ \cos A = \sin B. \] Step 3: Verify with triangle properties.
Check if the simplified equation holds under the cosine and sine laws, ensuring the result complies with the geometry of triangle ABC.