Find the modulus of `(1+i)/(1-i)-(1-i)/(1+i)`.

Explanation

To find the modulus of the expression \(\frac{1+i}{1-i} - \frac{1-i}{1+i}\), we can follow these steps:

### Step 1: Find a common denominator
We can combine the two fractions by finding a common denominator. The common denominator for \((1-i)\) and \((1+i)\) is \((1-i)(1+i)\).

\[ \frac{1+i}{1-i} - \frac{1-i}{1+i} = \frac{(1+i)(1+i) - (1-i)(1-i)}{(1-i)(1+i)} \]

### Step 2: Simplify the numerator
Now, we need to simplify the numerator:

1. Calculate \((1+i)(1+i)\):
\[ (1+i)(1+i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i \]

2. Calculate \((1-i)(1-i)\):
\[ (1-i)(1-i) = 1 - 2i + i^2 = 1 - 2i - 1 = -2i \]

Now, substituting these results back into the numerator:
\[ 2i - (-2i) = 2i + 2i = 4i \]

### Step 3: Simplify the denominator
Next, we simplify the denominator:
\[ (1-i)(1+i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

### Step 4: Combine the results
Now we can combine the results:
\[ \frac{4i}{2} = 2i \]

### Step 5: Find the modulus
Finally, we find the modulus of \(2i\):
\[ |2i| = 2 \]

### Final Answer
The modulus of \(\frac{1+i}{1-i} - \frac{1-i}{1+i}\) is \(2\).

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