IF $$A+B+C=\pi$$
then prove that-
$$cosA+cosB+cosC=1+4\sin \frac {A}{2}\sin \frac{B}{2} \sin \frac{C}{2}$$

Question

IF $$A+B+C=\pi$$
then prove that-
$$cosA+cosB+cosC=1+4\sin \frac {A}{2}\sin \frac{B}{2} \sin \frac{C}{2}$$

anonymous · Answer

$$\cos A+\cos B+\cos C=\cos A+\cos B+\cos (\pi-A-B)\=\cos A+\cos B-\cos (A+B)=\cos A+\cos B-\cos A \cos B+\sin A \sin B\=1-(1-\cos A)(1-\cos B)+\sin A \sin B \=1-4 \  \sin^2 \frac{A}{2}  \sin^2 \frac{B}{2}+4\sin \frac{A}{2}  \cos \frac{A}{2} \sin \frac{B}{2}  \cos \frac{B}{2}$$

now just factor out $$ 4 \ \sin \frac{A}{2}  \sin \frac{B}{2}$$
see what happens

anonymous · Answer

http://sg.answers.yahoo.com/question/index?qid=20091218205518AAtsWzf

anonymous · Answer

continuing the work of @mukushla  one gets

$$
1-4 \  \sin^2 \frac{A}{2}  \sin^2 \frac{B}{2}+4\sin \frac{A}{2}  \cos \frac{A}{2} \sin \frac{B}{2}  \cos \frac{B}{2}\
1+4 \  \sin  \frac{A}{2}  \sin \frac{B}{2}\left(-\sin  \frac{A}{2}  \sin \frac{B}{2}+  \cos \frac{A}{2}   \cos \frac{B}{2}
\right)\
1+4 \  \sin  \frac{A}{2}  \sin \frac{B}{2} \cos \left(\frac A 2 + \frac B 2\right)
\
1+4 \  \sin  \frac{A}{2}  \sin \frac{B}{2} \cos \left(\frac \pi 2-\frac C 2\right)
\
1+4 \  \sin  \frac{A}{2}  \sin \frac{B}{2} \sin\frac C 2 
\



$$