What is $\def\arcsec{\operatorname{arcsec}}\tan(\arcsec(x))$ simplified and why?
More specifically, I followed this reasoning, but apparently it is wrong:
$\tan(\arcsec(x))=\sqrt{\sec^2(\arcsec(x))-1}=\sqrt{x^2-1}$
What is wrong with this reasoning?
Apparently the answer is: $\sqrt{x^2-1}$ for $x\ge1$ and $-\sqrt{x^2-1}$ for $x\le1$ Why is this the right answer?
Best Answer
By definition we have: $$ y=\mbox{arcsec}x \iff \sec y=x \Rightarrow \cos y=\frac{1}{x}\Rightarrow \sin y=\sqrt{1-\frac{1}{x^2}} \Rightarrow $$
$$ \Rightarrow \tan y = \frac{\sin y}{\cos y}=x\sqrt{1-\frac{1}{x^2}}=\frac{x}{|x|}\sqrt{x^2-1} $$