$$\dfrac{\dfrac{x}{y}-\dfrac{y}{x}}{\dfrac{x}{y}+\dfrac{y}{x}}=\dfrac{x^2-y^2}{x^2+y^2}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\frac{x}{y}-\frac{y}{x}}{\frac{x}{y}+\frac{y}{x}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{x^2-y^2}{xy}}{\frac{x^2+y^2}{xy}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{x^2-y^2}{x^2+y^2}\end{aligned} $$ | |
| ① | Subtract $ \dfrac{y}{x} $ from $ \dfrac{x}{y} $ to get $ \dfrac{ \color{purple}{ x^2-y^2 } }{ xy }$. To subtract raitonal expressions, both fractions must have the same denominator. |
| ② | Add $ \dfrac{x}{y} $ and $ \dfrac{y}{x} $ to get $ \dfrac{ \color{purple}{ x^2+y^2 } }{ xy }$. To add raitonal expressions, both fractions must have the same denominator. |
| ③ | Divide $ \dfrac{x^2-y^2}{xy} $ by $ \dfrac{x^2+y^2}{xy} $ to get $ \dfrac{ x^2-y^2 }{ x^2+y^2 } $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Cancel $ \color{red}{ xy } $ in first and second fraction. Step 3: Multiply numerators and denominators. Step 4: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{x^2-y^2}{xy} }{ \frac{\color{blue}{x^2+y^2}}{\color{blue}{xy}} } & \xlongequal{\text{Step 1}} \frac{x^2-y^2}{xy} \cdot \frac{\color{blue}{xy}}{\color{blue}{x^2+y^2}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{x^2-y^2}{\color{red}{1}} \cdot \frac{\color{red}{1}}{x^2+y^2} \xlongequal{\text{Step 3}} \frac{ \left( x^2-y^2 \right) \cdot 1 }{ 1 \cdot \left( x^2+y^2 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ x^2-y^2 }{ x^2+y^2 } \end{aligned} $$ |