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Question Number 21268 by oyshi last updated on 18/Sep/17

i f A + B + C = π

s o p r o o f

\sin A + \sin B + \sin C = 4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}

Answered by myintkhaing last updated on 18/Sep/17

L . H . S = \sin A + \sin B + \sin C

= 2 \sin \frac{A + B}{2} \cos \frac{A - B}{2} + 2 \sin \frac{C}{2} \cos \frac{C}{2}

= 2 \sin (\frac{π}{2} - \frac{C}{2}) \cos (\frac{A - B}{2}) + 2 \sin (\frac{π}{2} - \frac{A + B}{2}) \cos \frac{C}{2}

= 2 \cos \frac{C}{2} \cos (\frac{A - B}{2}) + 2 \cos (\frac{A + B}{2}) \cos \frac{C}{2}

= 2 (\cos \frac{A + B}{2} + \cos \frac{A - B}{2}) \cos \frac{C}{2}

= 2 (2 \cos \frac{A}{2} \cos \frac{B}{2}) \cos \frac{C}{2}

You can't use 'macro parameter character #' in math mode

use

A + B = π - C \Rightarrow \frac{A + B}{2} = \frac{π}{2} - \frac{C}{2}

\sin x + \sin y = 2 \sin \frac{x + y}{2} \cos \frac{x - y}{2}

\cos x + \cos y = 2 \cos \frac{x + y}{2} \cos \frac{x - y}{2}