# How do you simplify \frac{\sec x}{\tan x}?

How do you simplify $\frac{\mathrm{sec}x}{\mathrm{tan}x}$?
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Bertha Jordan
Use $\mathrm{tan}x=\frac{\mathrm{sin}x}{\mathrm{cos}x}$
And $\mathrm{sec}x=\frac{1}{\mathrm{cos}x}$
So, $\frac{\mathrm{sec}x}{\mathrm{tan}x}=\frac{1}{\mathrm{cos}x}\cdot \frac{\mathrm{cos}x}{\mathrm{sin}x}=\frac{1}{\mathrm{sin}x}=\mathrm{csc}x$
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Navreaiw
Let's use the definitions of $\mathrm{sec}x$ and $\mathrm{tan}x$ to simplify this. We know
$\mathrm{sec}x=\frac{1}{\mathrm{cos}x}$ and $\mathrm{tan}x=\frac{\mathrm{sin}x}{\mathrm{cos}x}$. Let's plug these values into our original expression. we get
$\frac{\frac{1}{\mathrm{cos}x}}{\frac{\mathrm{sin}x}{\mathrm{cos}x}}=\frac{1}{\mathrm{sin}x}⇒\mathrm{csc}x$
Therefore, $\frac{\mathrm{sec}x}{\mathrm{tan}x}=\mathrm{csc}x$
Hope this helps!