1. The greatest value

Find the greatest value of E :
$\displaystyle E=xy+x\sqrt{y^{2}-1}+y\sqrt{1-x^{2}}-\sqrt{(1-x^{2})(1-y^{2})}$

2. First, identify any constraints on the variables. For example, any expression inside a square root has to be greater than or equal to 0.

$\displaystyle y^2 - 1 \geq 0$
$\displaystyle y^2 \geq 1$
$\displaystyle y \leq -1$ or $\displaystyle y \geq 1$

$\displaystyle 1 - x^2 \geq 0$
$\displaystyle -x^2 \geq -1$
$\displaystyle x^2 \leq 1$
$\displaystyle -1 \leq x \leq 1$

$\displaystyle (1 - x^2)(1 - y^2) \geq 0$
$\displaystyle -1 \leq x \leq 1$ and $\displaystyle -1 \leq y \leq 1$, or ($\displaystyle x \leq -1$ or $\displaystyle x \geq 1$) and ($\displaystyle y \leq -1$ or $\displaystyle y \geq 1$)

Let's examine $\displaystyle -1 \leq y \leq 1$. We know $\displaystyle y \leq -1$ or $\displaystyle y \geq 1$. Therefore, $\displaystyle -1 < y < 1$ is false so the only solutions are $\displaystyle y = -1$ or $\displaystyle y = 1$.

Now let's examine $\displaystyle x \leq -1$ or $\displaystyle x \geq 1$. We know $\displaystyle -1 \leq x \leq 1$. Therefore, $\displaystyle x < -1$ and $\displaystyle x > 1$ are false so the only solutions are $\displaystyle x = -1$ or $\displaystyle x = 1$.

Now we can replace $\displaystyle -1 \leq y \leq 1$ with $\displaystyle y = -1$ or $\displaystyle y = 1$ and $\displaystyle x \leq -1$ or $\displaystyle x \geq 1$ with $\displaystyle x = -1$ or $\displaystyle x = 1$ to produce:

$\displaystyle -1 \leq x \leq 1$ and ($\displaystyle y = -1$ or $\displaystyle y = 1$), or ($\displaystyle x = -1$ or $\displaystyle x = 1$) and ($\displaystyle y \leq -1$ or $\displaystyle y \geq 1$)

From here, we can replace y with -1 and determine for which value of x does the simplified expression have a maximum. Then, we can repeat the replace y with 1 and repeat the procedure. Finally, we can compare those results to replacing x with -1 and 1 and finding for which value of y does this simplified expression have a maximum.

If you're allowed to use calculus, then I know I much more simple method.