Prove the uniqueness of $\sin x_i=1$ (only) in this problem If 
$$\sin x_1+ \sin  x_2+ \sin  x_3=3 $$ 
then prove that  
$$\cos x_1+ \cos  x_2+ \cos x_3=0 $$
My try:- as simply we can say that $\sin x_1=1$ we can say that $\cos x_1=0$. So $\cos x_1+ \cos  x_2+ \cos x_3=0 $. But is there any better proof than this . What I mean is to prove the uniqueness of $\sin x_i=1$ (only).

Once $\sin x \le 1$ then 
$$\sin x_1+\sin x_2 + \sin x_3 \le 3$$
and the equality holds only if $\sin x_1=\sin x_2 = \sin x_3=1 \quad (1)$. 
Because if for some $i$ we get $\sin x_i <1 $ then:
$$\sin x_1+\sin x_2 + \sin x_3 < 3$$
From $(1)$ we then conclude that:
$$\cos x_1=\cos x_2 = \cos x_3=0$$
