Cannot understand trig manipultion $a$ is a radius in a origin centered circle.
Any point on a circle circumference has coordinates $(a\ \cos\theta, a \ \sin\theta)$. 
I do not understand the last transformation in this expression $x = \sqrt{a^2 - y^2} = \sqrt{a^2 - a^2 \sin^2\theta } = a \cos\theta $ 
How do we get $ \sqrt{a^2 - a^2 \sin^2\theta } = a \cos\theta $ ???
Note: this is just a part of calc lecture in MIT ocw

It's from the Pythagorean identity $\cos^2\theta + \sin^2\theta = 1$.
Multiply both sides by $a^2$.  Then $a^2\cos^2\theta + a^2\sin^2\theta = a^2$.  Now subtract $a^2\sin^2\theta$ from both sides.  Then take square roots.
