[Trigonometric Inequalities]

Find the maximum value of the expression $\sin x \sin y \sin z + \cos x \cos y \cos z$.
What is the greatest value of the expression $\sin x \sin y+\cos x \cos y$?

## Solution

When $x=y=z=0$, the expression $\sin x \sin y \sin z+\cos x \cos y \cos z$ equals 1. We will show that a value greater than 1 cannot be achieved. The expression $\sin x \sin y \sin z+\cos x \cos y \cos z_{\text {does not }}$ exceed

$|\sin x\|\sin y|+| \cos x\| \cos y| \cdot$ There exist angles $x^{\prime}, y^{\prime}$ from the interval $[0, \pi / 2]$ such that $\sin x^{\prime}=|\sin x|_{\text {and }} \sin y^{\prime}=|\sin y|$. Then $|\sin x\|\sin y|+| \cos x\| \cos y|=\sin x^{\prime} \sin y^{\prime}+\cos x^{\prime} \cos y^{\prime}=\cos \left(x^{\prime}-y^{\prime}\right)$, which obviously does not exceed 1.

## Answer

1.00