OK. Let me start.

The formula 

   (cos(x) + i*sin(x))*(cos(y)+ i*sin(y)) = cos(y+x) + i*sin(y+x)    (1)

is valid for any two complex numbers 

   u = (cos(x) + i*sin(x)) and v = (cos(y) + i*sin(y))               (2)

that have the modulus 1 (are the "unit" vectors of the length 1 in the unit circle in the complex domain).

The "x" and "y" are "arguments" of the complex numbers "u" and "v".
Geometrically, it means that "x" and "y" are the angles on the unit circle.

This formula and the complex numbers came from the section of the school math that is called "Complex numbers". 
So, I assume that you have the minimal introductory knowledge to understand what I am writing.

Actually, there is a block of my lessons on complex numbers in this site, that might be useful for you.

The list of these lessons is at the end of my post, and you can look into them later. Right now I will continue.

Let us perform this multiplication of the two complex numbers in the left side of (1). 
Using the rules for multiplication of complex numbers, you have

  (cos(x) + i*sin(x))*(cos(y)+ i*sin(y)) = 

= (cos(x)*cos(y) - sin(x)*sin(y))  +  i*(sin(x)*cos(y) + cos(x)*sin(y).   (3)
   ----------------------------         -----------------------------
    This is the real part                And this is the imaginary
       of the product                       part of the product

Now compare it with the right side of the formula (1).

The complex number in the left side of (1), which is (3) is equal to the complex number in the right side of (1).
It means that the real parts of (3) and (1) are equal, as well as the imaginary parts of (3) and (1) are equal too.

In other words,

   cos(x)*cos(y) - sin(x)*sin(y) = cos(x+y),    (4)
   sin(x)*cos(y) + cos(x)*sin(y) = sin(x+y).    (5)

So, based on the equality (1) for complex numbers, you got these formulas for angles "x" and"y"

   cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y)   and
   sin(x+y) = sin(x)*cos(y) + cos(x)*sin(y).

If you studied Trigonometry before, you can recognize that the last two formulas are the "addition formulas for cosine and sine".

That is all my story for now.

At the end, as I promised, get the list of my lessons on complex numbers in this site:

    - Complex numbers and arithmetic operations on them

    - Complex plane

    - Addition and subtraction of complex numbers in complex plane

    - Multiplication and division of complex numbers in complex plane

    - Raising a complex number to an integer power

    - How to take a root of a complex number

    - Solution of the quadratic equation with real coefficients on complex domain

    - How to take a square root of a complex number

    - Solution of the quadratic equation with complex coefficients on complex domain