Let $x = \mathrm{asec}\,(\theta)$ in $\sqrt{x^2-a^2}$. Find $\sin(\theta)$ and $\cos(\theta)$

Question

Let $x = a \sec\,(\theta)$ in $\sqrt{x^2-a^2}$.

If $a>0$ and $0 < \theta < \pi/2$, then find $\sin(\theta)$ and $\cos(\theta)$.

All help is appreciated.

Answer 1

If $x=a \sec \theta$ then $\cos \theta = a/x$ and $\sin \theta = \sqrt{1-a^2/x^2}$ since for $0 < \theta < \pi /2$ we have $\sin \theta = \sqrt{1-\cos^2 \theta}$.