How can you maximize a trigonometric expression?

[anemone]

Maximize $\sin x \cos y+\sin y \cos z+\sin z \cos x$ for all real $x,\,y$ and $z$.

 

[kaliprasad]

Maximize $\sin x \cos y+\sin y \cos z+\sin z \cos x$ for all real $x,\,y$ and $z$.

Spoiler

from cyclic symmetry it is maximum when $x = y =z$
we get expression = $\frac{3}{2} \sin 2x$ and when $x = y = z= \frac{\pi}{4}$(this is one of the values) it is maximum = $\frac{3}{2}$