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

To add 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 }$ and the second by $\color{blue}{ i }$.

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

$$ i^2 = -1 $$

Divide $ \, 1+2i \, $ by $ \, -1+i \, $ to get $\,\, \dfrac{1-3i}{2} $.
( view steps )

Add $ \dfrac{1-3i}{2} $ and $ \dfrac{1}{1-i} $ to get $ \dfrac{ \color{purple}{ 3i^2-4i+3 } }{ -2i+2 }$.

To add 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 }$ and the second by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{1-3i}{2} + \frac{1}{1-i} & = \frac{ \left( 1-3i \right) \cdot \color{blue}{ \left( -i+1 \right) }}{ 2 \cdot \color{blue}{ \left( -i+1 \right) }} + \frac{ 1 \cdot \color{blue}{ 2 }}{ \left( 1-i \right) \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ -i+1+3i^2-3i } }{ -2i+2 } + \frac{ \color{purple}{ 2 } }{ -2i+2 }=\frac{ \color{purple}{ 3i^2-4i+3 } }{ -2i+2 } \end{aligned} $$

$$ 3i^2 = 3 \cdot (-1) = -3 $$

Simplify numerator

$$ \, \color{blue}{ -\cancel{3}} \,-4i+ \, \color{blue}{ \cancel{3}} \, = -4i $$

Divide $ \, -4i \, $ by $ \, 2-2i \, $ to get $\,\, 1-i $.
( view steps )

Combine like terms:

$$ -i+1 = -i+1 $$