Question

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

Answer 1

Recall that

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

which means

`(\sin\theta)/(√(1-\sin^2\theta))=(\sin\theta)/(√(\cos^2\theta))`

Now,
`\cos\theta` could be positive or negative, which means
`√(\cos^2\theta)=|\cos\theta|`. If we specifically knew the sign of
`\cos\theta` was positive, then we can simplify and write

`(\sin\theta)/(√(1-\sin^2\theta))=(\sin\theta)/(\cos\theta)=\tan\theta`

or if it's negative,

`(\sin\theta)/(√(1-\sin^2\theta))=(\sin\theta)/(-\cos\theta)=-\tan\theta`