Question

If $A+B+C=\pi$, show that $\cos \left(A/2\right)+\cos \left(B/2\right)-\cos \left(C/2\right)$ $=4\cos \frac{\pi +A}{4}\cos \frac{\pi +B}{4}\cos \frac{\pi -C}{4}$.

Answer 1

L.H.S. $=2\cos \frac{A+B}{4}\cos \frac{A-B}{4}-\sin (\frac{\pi }{2}-\frac{C}{2})$.
$=2\cos \frac{\pi -C}{4}\cos \frac{A-B}{4}-2\sin \frac{\pi -C}{4}\cos \frac{\pi -C}{4}$
$=2\cos \frac{\pi -C}{4}[\cos \frac{A-B}{4}-\sin \frac{\pi -C}{4}]$
$=2\cos \frac{\pi -C}{4}[\cos \frac{A-B}{4}-\sin \left(\frac{A+B}{4}\right)]$
$=2\cos \frac{\pi -C}{4}[\cos \frac{A-B}{4}+\cos (\frac{\pi }{2}+\frac{A+B}{4})]$
$[\because \cos \{\left(\pi /2\right)+\theta \}=-\sin \theta ]$
$=2\cos \frac{\pi -C}{4}\left[2\cos \frac{\pi +A}{4}\cos \frac{\pi +B}{4}\right]$
$=4\cos \frac{\pi +A}{4}\cos \frac{\pi +B}{4}\cos \frac{\pi -C}{4}$.