$\sin x+\sin y+\sin z=4\cos\frac{x}{2}\cos\frac{y}{2}\cos\frac{z}{2}$

Question

If $x+y+z=\pi$ how to prove that $\sin x+\sin y+\sin z=4\cos\frac{x}{2}\cos\frac{y}{2}\cos\frac{z}{2}$?

I got that $\sin x+\sin y+\sin z=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}+\sin x\cos y+\sin y\cos x$ and I don't know what to do next.

Can somebody help me, please?

Answer 1

I hope you this helps with the question

=2*sin⁡((x+y)/2)* cos⁡((x-y)/2)+2*sin⁡(z/2)*cos⁡(z/2) =2*sin⁡((π-z)/2)* cos⁡((x-y)/2)+2*sin⁡((π-(x+y))/2) *cos⁡(z/2) =2*cos⁡(z/2)*cos⁡((x-y)/2)+2*cos⁡((x+y)/2)*cos⁡(z/2) =2*cos⁡(z/2)*[cos⁡((x-y)/2)+cos⁡((x+y)/2)] =2*cos⁡(z/2)*[2*cos⁡(x/2)*cos⁡(y/2)] =4*cos⁡(x/2)*cos⁡(y/2)*cos⁡(z/2)