simplify

**Raoh** · July 15th 2009, 02:16 PM

hello

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

**mr fantastic** · July 15th 2009, 03:07 PM

Originally Posted by Raoh

hello

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

Without further information, no it can't.

**Soroban** · July 15th 2009, 03:27 PM

Hello, Raoh!

Can $\arctan\!\left(\frac{1}{\sqrt{1-x^{2}}}\right)$ be simplified ?

Well, we can make it "look nicer" . . . but that's all.

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

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

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

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

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

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

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