Question

Simplify sin θ / square root 1 - sin^2 θ

Answer 1

Recall that

$\cos^2\theta+\sin^2\theta=1$

which means

$\dfrac{\sin\theta}{\sqrt{1-\sin^2\theta}}=\dfrac{\sin\theta}{\sqrt{\cos^2\theta}}$

Now, $\cos\theta$ could be positive or negative, which means $\sqrt{\cos^2\theta}=|\cos\theta|$. If we specifically knew the sign of $\cos\theta$ was positive, then we can simplify and write

$\dfrac{\sin\theta}{\sqrt{1-\sin^2\theta}}=\dfrac{\sin\theta}{\cos\theta}=\tan\theta$

or if it's negative,

$\dfrac{\sin\theta}{\sqrt{1-\sin^2\theta}}=\dfrac{\sin\theta}{-\cos\theta}=-\tan\theta$