The angular momentum of a planet of mass M moving around the sun in...

For small displacement $ds$ of the planet its area can be written as

$dA = \dfrac{1}{2}rdl$

$= \dfrac{1}{2}rdsin\theta$

A.vel $= \dfrac{dA}{dt} = \dfrac{1}{2}rsin\theta\dfrac{ds}{dt} = \dfrac{Vrsin\theta}{2}$

$\dfrac{dA}{dt} = \dfrac{1}{2}\dfrac{mVrsin\theta}{m} = \dfrac{L}{2m}$