1. ## The greatest value

Find the greatest value of E :
$
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.

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

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

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

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

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

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

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

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.