Simplify the following expression: (1 + 1)/((1+i)ⁿ - 1)(1+i)ⁿ - 1

The expression from the image is:

$$\frac{1 + \frac{1}{(1+i)^n - 1}}{(1+i)^n - 1}$$

The numerator is $1 + \frac{1}{(1+i)^n - 1}$. To combine these terms, find a common denominator, which is $(1+i)^n - 1$.

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

Combine the numerators:

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

Now the expression becomes:

$$\frac{\frac{(1+i)^n}{(1+i)^n - 1}}{(1+i)^n - 1}$$

To simplify a complex fraction $\frac{\frac{A}{B}}{C}$, we can rewrite it as $\frac{A}{B} \times \frac{1}{C} = \frac{A}{BC}$.
In this case, $A = (1+i)^n$, $B = (1+i)^n - 1$, and $C = (1+i)^n - 1$.

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

Multiply the numerators and the denominators:

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