Solve Algebra Expression: \frac{1}{\sqrt{\left( \frac{y}{x}\right)^2 +1}}

[FrogPad]

I'm working with this expression and I do not understand how to simplify it by hand:

$$\frac{1}{\sqrt{\left( \frac{y}{x}\right)^2 +1}}$$

My TI-89 reduces it to:
$$\frac{|x|}{\sqrt{x^2+y^2}}$$

How is it doing this? This is not homework. I'm sure it would be acceptiable to just put the simplification down on paper... but if you would rather give hints, that's fine. Thanks :)

The original expression was taken from:
$$\cos \tan^{-1} \frac{y}{x}$$

 

[assyrian_77]

Under the squareroot you have the expression $$\frac{x^2}{y^2}+1$$

Remember that
$$\frac{a}{b}\pm\frac{c}{d}=\frac{ad\pm bc}{bd}$$

And also, remember that
$$\frac{1}{\frac{a}{b}}=\frac{b}{a}$$

Do you see know how your calculator did it?

 

[FrogPad]

:)

hehe

god my algebra is WEAK.
Thanks.

 

[HallsofIvy]

Or: multiply both numerator and denominator by |x|:
$$\frac{|x|}{|x|\sqrt{\frac{y^2}{x^2}+ 1}}= \frac{|x|}{\sqrt{x^2(\frac{y^2}{x^2+ 1)}}= \frac{|x|}{\sqrt{y^2+ x^2}}$$