Question

If \(A, B, C\) are the angles of a triangle, then \(\cos B + \cos C - \cos A + 1\) is equal to

• A. \(4\sin\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}\)
• B. \(-4\sin\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}\)
• C. \(4\cos\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\)
• D. \(4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\)

Hint:

Always use \(A+B+C=\pi\) in triangle trig problems.

Answer 1

The Correct Option is A

Concept: \[ A+B+C=\pi \]

Step 1: Use identity.
\[ \cos B + \cos C = 2\cos\frac{B+C}{2}\cos\frac{B-C}{2} \] \[ = 2\cos\frac{\pi-A}{2}\cos\frac{B-C}{2} = 2\sin\frac{A}{2}\cos\frac{B-C}{2} \]

Step 2: Combine terms.
Using identities and simplification: \[ = 4\sin\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2} \]