01 Rectangle of maximum perimeter inscribed in a circle

Perimeter of rectangle,
$P = 2x + 2y$
 

Where:
$x = D \cos \theta$

$y = D \sin \theta$
 

$P = 2D \cos \theta + 2D \sin \theta$

$\dfrac{dP}{d\theta} = -2D \sin \theta + 2D \cos \theta = 0$

$-\sin \theta + \cos \theta = 0$

$\sin \theta = \cos \theta$

$\dfrac{\sin \theta}{\cos \theta} = 1$

$\tan \theta = 1$

$\theta = 45^\circ$
 

$x = D \cos 45^\circ = 0.707D$

$y = D \sin 45^\circ = 0.707D$
 

$x = y \, \text{ (square)}$           answer
 

See also the solution using algebraic function.