Question

If \(A+B+C=\pi\), then \[ 3-2\left( \cos\frac{A}{2}\cos\frac{B}{2}\sin\frac{C}{2} +\cos\frac{A}{2}\sin\frac{B}{2}\cos\frac{C}{2} +\sin\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2} \right)= \]

• A. \(\sin^2A+\sin^2B+\sin^2C\)
• B. \(\cos^2A+\cos^2B+\cos^2C\)
• C. \(\sin^2\dfrac{A}{2}+\sin^2\dfrac{B}{2}+\sin^2\dfrac{C}{2}\)
• D. \(\cos^2\dfrac{A}{2}+\cos^2\dfrac{B}{2}+\cos^2\dfrac{C}{2}\)

Hint:

Whenever \(A+B+C=\pi\), convert complicated trigonometric expressions into half-angle identities for easier simplification.

Answer 1

The Correct Option is C

Concept: For angles of a triangle, \[ A+B+C=\pi \] Several half-angle identities become useful for simplification.

Step 1: Use the identity for sum of half-angle products.
Given expression: \[ 3-2\left( \cos\frac{A}{2}\cos\frac{B}{2}\sin\frac{C}{2} +\cos\frac{A}{2}\sin\frac{B}{2}\cos\frac{C}{2} +\sin\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2} \right) \] Since \[ A+B+C=\pi \] we use \[ \cos\frac{A}{2}\cos\frac{B}{2}\sin\frac{C}{2} +\cos\frac{A}{2}\sin\frac{B}{2}\cos\frac{C}{2} +\sin\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2} = \cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2} \] Thus expression becomes \[ 3-2\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2} \]

Step 2: Apply half-angle relations.
Using standard triangle identities, \[ 3-2\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2} = \sin^2\frac{A}{2} +\sin^2\frac{B}{2} +\sin^2\frac{C}{2} \] Hence, \[ \boxed{ \sin^2\frac{A}{2} +\sin^2\frac{B}{2} +\sin^2\frac{C}{2} } \]