1. ## simplify

hello
Can $\displaystyle arctan(\frac{1}{\sqrt{1-x^{2}}})$ be simplified ?
thanks

2. Originally Posted by Raoh
hello
Can $\displaystyle arctan(\frac{1}{\sqrt{1-x^{2}}})$ be simplified ?
thanks
Without further information, no it can't.

3. Hello, Raoh!

Can $\displaystyle \arctan\!\left(\frac{1}{\sqrt{1-x^{2}}}\right)$ be simplified ?
Well, we can make it "look nicer" . . . but that's all.

We have: .$\displaystyle \theta \:=\:\arctan\left(\frac{1}{\sqrt{1-x^2}}\right)$

. . Hence: .$\displaystyle \tan\theta \:=\:\frac{1}{\sqrt{1-x^2}} \:=\:\frac{opp}{adj}$

So $\displaystyle \theta$ is in a right triangle with: .$\displaystyle opp = 1,\;adj = \sqrt{1-x^2}$

Using Pythagorus, we have: .$\displaystyle hyp \,=\,\sqrt{2-x^2}$

Then: .$\displaystyle \csc\theta \:=\:\frac{hyp}{opp} \:=\:\frac{\sqrt{2-x^2}}{1} \quad\Rightarrow\quad \theta \:=\:\text{arccsc}\left(\sqrt{2-x^2}\right)$

Therefore: .$\displaystyle \arctan\!\left(\frac{1}{\sqrt{1-x^2}}\right) \;=\;\text{arccsc}\left(\sqrt{2-x^2}\right)$