Kaila Bray

2022-12-31

How to simplify $\frac{\mathrm{sin}\left(-x\right)}{\mathrm{cos}\left(-x\right)}$?

Shea Pace

Expert

$\mathrm{cos}x$ is an even function. It follows that $\mathrm{cos}\left(-x\right)=\mathrm{cos}x$
$\mathrm{sin}x$ is an odd function. It follows that $\mathrm{sin}\left(-x\right)=-\mathrm{sin}x$
$\frac{\mathrm{sin}\left(-x\right)}{\mathrm{cos}\left(-x\right)}=-\frac{\mathrm{sin}x}{\mathrm{cos}x}=-\mathrm{tan}x$

Demetrius Arnold

Expert

It is an even function, so $\mathrm{cos}\left(-x\right)=\mathrm{cos}\left(x\right)$
Similarly, because it is an odd function, $\mathrm{sin}\left(-x\right)=-\mathrm{sin}\left(x\right)$
So, the above ratio can be written
$-\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$ which is equal to $-\mathrm{tan}\left(x\right)$

Do you have a similar question?