Example 8

Suppose that a, b, c, , and are real numbers, with the following conditions: , , k is any integer.

	(1)
	(2)

Prove that .

You might already notice that this is the hardest problem in our entire Learning section. Yes, it's a difficult problem. Especially you might not even know where you should start.

Like the previous one, this is a conditional identity too. We see this problem gives us many more conditions. How do we start? What formulas can help us to prove it? We believe that every problem has some clues. Which tell us how to solve it. Let's see what clues this problem has.

From the left side of the goal identity, we see the angle , which let us believe it must be relate to the Sum to Product formulas. So let's try to add equations (1) and (2) by using the Sum to Product formulas. We'll get the angle and see if we can get anything else.

(1) + (2)

(3)

From (3), we use the Sum to Product formulas to get:

Then we divide 2 from both sides and take out the common factor:

(4)

Let's look at equation (4), because the goal identity has square at both sides, so if we want to get we need square both sides of equation (4).

(5)