Easy Equations

$\begin{aligned} x=\sqrt{\frac{y-1}{y+1}} &\Longleftrightarrow x^2 = \frac{y-1}{y+1}\\[1em] &\Longleftrightarrow (y+1)x^2 = y-1\\[1em] &\Longleftrightarrow yx^2+x^2=y-1\\[1em] &\Longleftrightarrow yx^2-y=-(x^2+1)\\[1em] &\Longleftrightarrow y(x^2-1)=-(x^2+1)\\[1em] &\Longleftrightarrow y = \frac{-1}{-1}\cdot \frac{x^2+1}{1-x^2}\\[1em] &\Longleftrightarrow y = \frac{x^2+1}{1-x^2} \end{aligned}$

$\begin{aligned}x &= \sqrt{\dfrac{y - 1}{y + 1}}\\\\x^2 &= \dfrac{y - 1}{y + 1}\\\\x^2&=\dfrac{y + 1 - 2}{y + 1}\\\\x^2 &= \dfrac{y + 1}{y + 1} - \dfrac{2}{y + 1}\\\\x^2&=1 - \dfrac{2}{y + 1}\\\\x^2 - 1 &=-\dfrac{2}{y + 1} \\\\\dfrac{2}{y + 1}&=1 - x^2\\\\\dfrac{2}{1 - x^2} &= y + 1\\\\\dfrac{2}{1 - x^2} - 1&= y \\\\\dfrac{2}{1 - x^2} - \dfrac{1 - x^2}{1 - x^2}&= y\\\\\dfrac{2 -(1 - x^2)}{1 - x^2}&= y\\\\\dfrac{1 + x^2}{1 - x^2}&= y\end{aligned}$

$\begin{aligned} x & = & \sqrt { \frac { y-1 }{ y+1 } } \\ { x }^{ 2 } & = & \frac { y-1 }{ y+1 } \\ (y+1){ x }^{ 2 } & = & y-1 \\ { x }^{ 2 }y+{ x }^{ 2 } & = & y-1 \\ y-{ x }^{ 2 }y & = & 1+{ x }^{ 2 } \\ y(1-{ x }^{ 2 }) & = & { 1+x }^{ 2 } \\ y & = & \boxed { \frac { 1+{ x }^{ 2 } }{ 1-{ x }^{ 2 } } } \end{aligned}$

First square both sides. Then the equation will be, x^2 = y - 1 / y + 1.

x^2 + 1 / x^2 - 1 = y - 1 + y + 1 / y - 1-y - 1. [Componendo - Dividendo].

y = 1 + x^2/1 - x^2