Let z and w be complex numbers such that |z| = |w| = 1, and zw is not equal to -1. Prove that
(z + w)/(zw + 1)
is a real number.
To prove that the expression (z + w)/(zw + 1) is a real number, we need to show that the imaginary part of the expression is equal to zero.
Let's start by calculating the imaginary part of (z + w)/(zw + 1):
Let z = a + bi and w = c + di, where a, b, c, and d are real numbers.
The imaginary part of (z + w) is b + d.
The imaginary part of (zw + 1) is (ad + bc) + (bd - ac)i.
Now, let's calculate the imaginary part of (z + w)/(zw + 1):
Im[(z + w)/(zw + 1)] = (b + d)/[(ad + bc) + (bd - ac)i] = (b + d)(ad + bc)/(ad + bc)^2 + (bd - ac)^2.
We want to show that this expression is equal to zero, so we need to prove that (b + d)(ad + bc) = 0.
Since |z| = |w| = 1, it implies that a^2 + b^2 = c^2 + d^2 = 1.
We know that zw is not equal to -1, which means that ad + bc is not equal to -1.
Therefore, we have two cases to consider:
Case 1: b + d = 0
In this case, the imaginary part of (z + w)/(zw + 1) is equal to zero.
Case 2: ad + bc = 0
In this case, the imaginary part of (z + w)/(zw + 1) is also equal to zero.
Thus, we have proven that the expression (z + w)/(zw + 1) is a real number.