Question

How is \[ \frac{1+\frac{x}{\sqrt{x^2+1}}}{x+\sqrt{x^2 + 1}} = \frac{1}{\sqrt{x^2 + 1}} \]
What techniques are used here?

Answer 1

Since numerator of LHS isn't that clear, it's:

\[ 1 + \frac{x}{\sqrt{x^2+1}} \]

Answer 2

Have you tried rationalizing the denominator? multiplying the top and bottom of the LHS fraction by \[x-\sqrt{x^2+1}\]?

Answer 3

nvm, thats actually the long way, there is a shorter way >.> just change:

Answer 4

\[1+\frac{x}{\sqrt{x^2+1}}=\frac{\sqrt{x^2+1}}{\sqrt{x^2+1}}+\frac{x}{\sqrt{x^2+1}}\]\[=\frac{x+\sqrt{x^2+1}}{\sqrt{x^2+1}}\]