Divide $ \, -i \, $ by $ \, 1+i \, $ to get $\,\, \dfrac{-1-i}{2} $.
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$$ \left( \color{blue}{i-1}\right) \cdot i = i^2-i $$

$$ i^2 = -1 $$

Subtract $ \dfrac{-1-i}{2} $ from $ 1 $ to get $ \dfrac{ \color{purple}{ i+3 } }{ 2 }$.

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 $\color{blue}{ 2 }$.

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

Add $ \dfrac{i+3}{2} $ and $ -1-i $ to get $ \dfrac{ \color{purple}{ -i+1 } }{ 2 }$.

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

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

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