Question

Make x the subject of the formula y = ${\displaystyle \frac{1-x^2}{1+x^2}}$. Find x, when y = ${\displaystyle \frac{1}{2}}$

Answer 1

y = ${\displaystyle \frac{1-x^2}{1+x^2}}$

⇒ y(1 + x²) = 1 - x²
⇒ y + yx² = 1 - x²
⇒ yx² + x² = 1 - y
⇒ x²( 1 + y) = 1 - y

⇒ x² = ${\displaystyle \frac{1-y}{1+y}}$

⇒ x = ${\displaystyle \sqrt{\frac{1-y}{1+y}}}$

Substituting y = ${\displaystyle \frac{1}{2}}$, we get

x = ${\displaystyle \sqrt{\frac{1-\frac{1}{2}}{1+\frac{1}{2}}}}$

= ${\displaystyle \sqrt{\frac{1}{3}}}$.