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Question

Answer

$\frac{i}{i + 1} - (1 - i)^{2} = \frac{5 i + 1}{2}$

Explanation

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$\frac{i}{i + 1} - (1 - i)^{2} \overset{◯}{1} \frac{i}{i + 1} - (1 - 2 i + i^{2}) \overset{◯}{2} \frac{i}{i + 1} - (1 - 2 i - 1) \overset{◯}{3} \frac{i}{i + 1} - - 2 i \overset{◯}{4} \frac{1 + i}{2} - - 2 i \overset{◯}{5} \frac{5 i + 1}{2}$
①	Find $(1 - i)^{2}$ using formula. $(A - B)^{2} = A^{2} - 2 \cdot A \cdot B + B^{2}$ where $A = 1$ and $B = i$ . $(1 - i)^{2} = 1^{2} - 2 \cdot 1 \cdot i + i^{2} = 1 - 2 i + i^{2}$
②	$i^{2} = - 1$
③	Combine like terms: $1 - 2 i - 1 = - 2 i$
④	Divide $i$ by $1 + i$ to get $\frac{1 + i}{2}$ . ( view steps )
⑤	Subtract $- 2 i$ from $\frac{1 + i}{2}$ to get $\frac{5 i + 1}{2}$ . Step 1: Write $- 2 i$ as a fraction by putting $1$ 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 second fraction by $2$ . $\frac{1 + i}{2} - - 2 i Step 1 \frac{1 + i}{2} - \frac{- 2 i}{1} = \frac{1 + i}{2} - \frac{( - 2 i ) \cdot 2}{1 \cdot 2} = Step 2 \frac{1 + i}{2} - \frac{- 4 i}{2} = \frac{5 i + 1}{2}$

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