# What is tan &#x2061;<!-- ⁡ --> (

What is$\mathrm{tan}\left(\mathrm{arcsec}\left(x\right)\right)$?
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Sawyer Day
By definition we have:
$y=\text{arcsec}x\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\mathrm{sec}y=x⇒\mathrm{cos}y=\frac{1}{x}⇒\mathrm{sin}y=\sqrt{1-\frac{1}{{x}^{2}}}⇒$
$⇒\mathrm{tan}y=\frac{\mathrm{sin}y}{\mathrm{cos}y}=x\sqrt{1-\frac{1}{{x}^{2}}}=\frac{x}{|x|}\sqrt{{x}^{2}-1}$