We have the expression:

$\frac{1}{1+i} - \frac{1}{1-i}$

To simplify this, let's first rewrite each term using a common technique: multiplying both the numerator and denominator by the conjugate of the denominator to eliminate the imaginary parts.

Step 1: Simplify $\frac{1}{1+i}$

Multiply both the numerator and denominator by the conjugate of $1+i$, which is $1-i$:

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

So, $\frac{1}{1+i} = \frac{1-i}{2}$.

Step 2: Simplify $\frac{1}{1-i}$

Similarly, multiply both the numerator and denominator by the conjugate of $1-i$, which is $1+i$:

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

So, $\frac{1}{1-i} = \frac{1+i}{2}$.

Step 3: Subtract the two fractions

Now we subtract the two results:

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

Final Answer:

$\frac{1}{1+i} - \frac{1}{1-i} = -i$

Would you like more details or have any questions?

Related Questions:

1. How do you multiply by the conjugate of a complex number?
2. Can this technique be used for simplifying complex fractions in general?
3. What are the steps for adding and subtracting complex numbers?
4. How do you simplify expressions involving imaginary numbers?
5. What are other methods to rationalize complex fractions?

Tip: When dealing with complex numbers in fractions, using the conjugate to remove the imaginary part from the denominator is a key technique.