# Inequality involving sum of four $\sin$

How would one show that, given $x,y,z,t\in[0,\pi]$, $$\sin x+\sin y+\sin z+\sin t\leq4\sin\left(\frac{x+y+z+t}{4}\right) ?$$ I recognize that $\frac{x+y+z+t}{4}$ is the arithmetic mean of $x,y,z,t$ but I don't see how to prove the inequality. Expressing the $\sin$ as integrals of $\cos$ seems to lead nowhere... Just a hint would be appreciated.

You could start with proving $\sin x +\sin y\le 2\sin \left(\frac {x+y}2\right)$ in the relevant range, which on expanding the left-hand side using the double angle formula, and the right-hand side using the formula for $\sin (a+b)$ gives $$\left(\sin \frac x2-\sin \frac y2\right)\left(\cos \frac x2-\cos \frac y2\right)\le 0$$
You can easily show the fact that $\sin w$ is a concave function if you know calculus, because its second derivative on your interval is always less than or equal to $0$, (the second derivative is $-\sin w$)