A rectangle is inscribed in a unit circle.  Find the largest possible area of the rectangle.
Let the dimensions of the rectangle be $a$ and $b.$  Since $a$ and $b$ are the legs of a triangle with hypotenuse 2, $a^2 + b^2 = 4.$

[asy]
unitsize (2 cm);

pair A, B, C, D;

A = dir(35);
B = dir(180 - 35);
C = dir(180 + 35);
D = dir(360 - 35);

draw(Circle((0,0),1));
draw(A--B--C--D--cycle);
draw(A--C);
draw(rightanglemark(C,D,A,4));

label("$a$", (A + D)/2, W);
label("$b$", (C + D)/2, N);

dot((0,0));
[/asy]

Then by AM-GM,
\[4 = a^2 + b^2 \ge 2ab,\]so $ab \le 2.$

Equality occurs when $a = b = \sqrt{2},$ so the largest possible area is $\boxed{2}.$