Simplify : `(1+2i)/(1-2i)-(1-2i)/(1+2i)`

Explanation

To simplify the expression \((1 + 2i)/(1 - 2i) - (1 - 2i)/(1 + 2i)\), we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \frac{1 + 2i}{1 - 2i} - \frac{1 - 2i}{1 + 2i} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \((1 - 2i)(1 + 2i)\). So we rewrite the expression:
\[ \frac{(1 + 2i)(1 + 2i) - (1 - 2i)(1 - 2i)}{(1 - 2i)(1 + 2i)} \]

### Step 3: Expand the numerators
Now we expand the numerators:
1. For \((1 + 2i)(1 + 2i)\):
\[ (1 + 2i)^2 = 1^2 + 2 \cdot 1 \cdot 2i + (2i)^2 = 1 + 4i - 4 = -3 + 4i \]
2. For \((1 - 2i)(1 - 2i)\):
\[ (1 - 2i)^2 = 1^2 - 2 \cdot 1 \cdot 2i + (-2i)^2 = 1 - 4i - 4 = -3 - 4i \]

### Step 4: Combine the results
Now we substitute back into the expression:
\[ \frac{(-3 + 4i) - (-3 - 4i)}{(1 - 2i)(1 + 2i)} \]
This simplifies to:
\[ \frac{-3 + 4i + 3 + 4i}{(1 - 2i)(1 + 2i)} = \frac{8i}{(1 - 2i)(1 + 2i)} \]

### Step 5: Simplify the denominator
Now we simplify the denominator:
\[ (1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - (-4) = 1 + 4 = 5 \]

### Step 6: Final expression
Thus, we have:
\[ \frac{8i}{5} \]
This can be written as:
\[ 0 + \frac{8}{5}i \]

### Final Answer
The simplified expression is:
\[ \frac{8}{5}i \]

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