The vertices of rectangle $PQRS$ lie on the curve defined by

Given that $PQ,RS$ are parallel to the $y$ axis, and $PS,QR$ are parallel the the $x$ axis, find the maximum possible area for the rectangle.

Let's say $A$ has coordinates $(\sin (2\theta ),\cos (\theta ))$, and so $B$ is of the form $(-\sin (2\theta ),\cos (\theta ))$, and so on. This should enable us to get a formula for the area, $A$, of the rectangle in terms of $\theta$.

To find the maximum value for the area, we need to think about

Don't try to find the value of $\theta$. You can instead infer the exact values of $\sin \theta$ and $\cos \theta$ by considering $\frac{dA}{d\theta }=0$. Once you know these, you can put them into the formula for the area to get its exact value.

In fully simplified form, the area of the rectangle can be given as

Give the value of