Foemicofeduz

2023-01-01

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

Anabel Phillips

$\frac{\mathrm{csc}\left(-x\right)}{\mathrm{cot}\left(-x\right)}=-\frac{\mathrm{csc}x}{-\mathrm{cot}x}=\frac{\mathrm{csc}x}{\mathrm{cot}x}=\frac{\frac{1}{\mathrm{sin}x}}{\frac{\mathrm{cos}x}{\mathrm{sin}x}}$
$=\frac{1}{\mathrm{cos}x}=\mathrm{sec}x$

Taniyah Hartman

Let: $\frac{\mathrm{csc}\left(-x\right)}{\mathrm{cot}\left(-x\right)}$
Let's apply the fact that $\mathrm{csc}\left(x\right)$ and $\mathrm{cot}\left(x\right)$ are odd functions:
$=\frac{-\mathrm{csc}\left(x\right)}{-\mathrm{cot}\left(x\right)}$
$=\frac{\mathrm{csc}\left(x\right)}{\mathrm{cot}\left(x\right)}$
Applying two common trigonometric identities will follow.; $\mathrm{csc}\left(x\right)=\frac{1}{\mathrm{sin}\left(x\right)}$ and $\mathrm{cot}\left(x\right)=\frac{\mathrm{cos}\left(x\right)}{\mathrm{sin}\left(x\right)}$:
$=\frac{\frac{1}{\mathrm{sin}\left(x\right)}}{\frac{\mathrm{cos}\left(x\right)}{\mathrm{sin}\left(x\right)}}$
$=\frac{1}{\mathrm{sin}\left(x\right)}\cdot \frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$
$=\frac{1}{\mathrm{cos}\left(x\right)}$
Finally, let's apply another standard trigonometric identity; $\mathrm{sec}\left(x\right)=\frac{1}{\mathrm{cos}\left(x\right)}$:
$=\mathrm{sec}\left(x\right)$

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