Given:
|a + ib| = 1

To prove:
1 + b * ai / 1 - b * ai = b * ai

Proof:

Step 1: Simplify the left-hand side of the equation

Let's start by simplifying the left-hand side of the equation:

1 + b * ai / 1 - b * ai

To simplify this expression, we can use the formula for the division of complex numbers:

(a + ib) / (c + id) = (ac + bd) / (c^2 + d^2) + (bc - ad)i / (c^2 + d^2)

In our case, a = 1, b = ai, c = 1, and d = -ai. Substituting these values into the formula, we get:

(1 + b * ai)/(1 - b * ai) = ((1 * 1) + (ai * -ai))/(1^2 + (-ai)^2) + ((ai * 1) - (1 * -ai)i)/(1^2 + (-ai)^2)

Simplifying further:

(1 - a^2i^2)/(1 + a^2i^2) + (a - ai^2)/(1 + a^2i^2)

Since i^2 = -1:

(1 - (-a^2))/(1 + a^2) + (a + a)/(1 + a^2)

Simplifying again:

(1 + a^2)/(1 + a^2) + (2a)/(1 + a^2)

The terms (1 + a^2) cancel out:

1 + (2a)/(1 + a^2)

Step 2: Simplify the right-hand side of the equation

Now let's simplify the right-hand side of the equation:

b * ai

Since multiplication of complex numbers is commutative, we can write this as:

ai * b

Using the formula for multiplying complex numbers:

(a + ib)(c + id) = (ac - bd) + (ad + bc)i

In our case, a = 0, b = ai, c = 1, and d = b. Substituting these values into the formula, we get:

(ai * 1) + (0 * b)i

Simplifying:

ai + 0i

Which is equal to:

ai

Step 3: Compare the simplified expressions

We have simplified the left-hand side of the equation to be:

1 + (2a)/(1 + a^2)

And the right-hand side of the equation is:

ai

To prove that the two sides are equal, we need to show that:

1 + (2a)/(1 + a^2) = ai

Multiplying both sides of the equation by (1 + a^2):

(1 + a^2) + 2a = ai(1 + a^2)

Expanding:

1 + a^2 + 2a = ai + ai^3

Since i^2 = -1 and i^3 = -i:

1 + a^2 + 2a = ai - ai

The terms ai and -ai cancel out, and we are left with:

1 + a