Subtract $ \dfrac{1}{1+i} $ from $ 1 $ to get $ \dfrac{ \color{purple}{ i } }{ i+1 }$.

Step 1: Write $ 1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ i+1 }$.

$$ \begin{aligned} 1- \frac{1}{1+i} & \xlongequal{\text{Step 1}} \frac{1}{\color{red}{1}} - \frac{1}{1+i} = \frac{ 1 \cdot \color{blue}{ \left( i+1 \right) }}{ 1 \cdot \color{blue}{ \left( i+1 \right) }} - \frac{ 1 }{ 1+i } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ i+1 } }{ i+1 } - \frac{ \color{purple}{ 1 } }{ i+1 }=\frac{ \color{purple}{ i } }{ i+1 } \end{aligned} $$

Divide $ \, i \, $ by $ \, 1+i \, $ to get $\,\, \dfrac{1+i}{2} $.
( view steps )

Divide $1$ by $ \dfrac{1+i}{2} $ to get $ \dfrac{2}{i+1} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Write $ 1 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{ \frac{\color{blue}{1+i}}{\color{blue}{2}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{2}}{\color{blue}{1+i}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{2}{1+i} \xlongequal{\text{Step 3}} \frac{ 1 \cdot 2 }{ 1 \cdot \left( 1+i \right) } \xlongequal{\text{Step 4}} \frac{ 2 }{ 1+i } = \\[1ex] &= \frac{2}{i+1} \end{aligned} $$

Subtract $ \dfrac{2}{i+1} $ from $ 1 $ to get $ \dfrac{ \color{purple}{ i-1 } }{ i+1 }$.

Step 1: Write $ 1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ i+1 }$.

$$ \begin{aligned} 1- \frac{2}{i+1} & \xlongequal{\text{Step 1}} \frac{1}{\color{red}{1}} - \frac{2}{i+1} = \frac{ 1 \cdot \color{blue}{ \left( i+1 \right) }}{ 1 \cdot \color{blue}{ \left( i+1 \right) }} - \frac{ 2 }{ i+1 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ i+1 } }{ i+1 } - \frac{ \color{purple}{ 2 } }{ i+1 }=\frac{ \color{purple}{ i-1 } }{ i+1 } \end{aligned} $$

Divide $ \, -1+i \, $ by $ \, 1+i \, $ to get $\,\, i $.
( view steps )