$$ i^{11} = i^{4 \cdot 2 + 3} = \left( i^4 \right)^{ 2 } \cdot i^3 = 1^{ 2 } \cdot (-i) = -i = -i $$$$ i^{21} = i^{4 \cdot 5 + 1} = \left( i^4 \right)^{ 5 } \cdot i^1 = 1^{ 5 } \cdot i = i $$

Subtract $ \dfrac{1}{i} $ from $ \dfrac{1}{-i} $ to get $ \dfrac{ \color{purple}{ 2i } }{ -i^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 }$ and the second by $\color{blue}{ -i }$.

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

$$ -i^2 = -(-1) = 1 $$