To simplify the expression $\frac{2}{1+i} - \frac{2}{1-i}$, we will follow these steps:

Step 1: Multiply by the conjugate

For each term, multiply both the numerator and denominator by the conjugate of the denominator. The conjugates are $1-i$ for $1+i$, and $1+i$ for $1-i$.

First term:

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

Second term:

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

Step 2: Simplify the denominators

The denominator for both terms becomes: $(1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 2$ So, both denominators become 2.

First term:

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

Second term:

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

Step 3: Subtract the two terms

Now subtract the two simplified terms: $(1 - i) - (1 + i) = 1 - i - 1 - i = -2i$

Thus, the simplified expression is: $\boxed{-2i}$

Would you like further details or have any questions?

Here are 5 related questions:

1. How can you multiply complex numbers using conjugates?
2. What is the meaning of the conjugate in complex numbers?
3. How do you add or subtract complex numbers?
4. How do you divide complex numbers using the conjugate?
5. Why is multiplying by the conjugate useful in simplifying fractions with complex numbers?

Tip: When dividing complex numbers, multiplying by the conjugate helps eliminate the imaginary part in the denominator.