Derivation of sum and difference identities for sine and cosine

John Kerl

January 2, 2012

The authors of your trigonometry textbook give a geometric derivation of the sum and difference identities for sine and cosine. I find this argument unwieldy — I don’t expect you to remember it; in fact, I don’t remember it. There’s a standard algebraic derivation which is far simpler. The only catch is that you need to use complex arithmetic, which we don’t cover in Math 111. Nonetheless, I will present the derivation so that you will have seen how simple the truth can be, and so that you may come to understand it after you’ve had a few more math courses. And in fact, all you need are the following facts:

• Complex numbers are of the form a+bi, where a and b are real numbers and i is defined to be a square root of −1. That is, i^{2} = −1. (Of course, (−i)^{2} = −1 as well, so −i is the other square root of −1.)

• The number a is called the real part of a + bi; the number b is called the imaginary part of a + bi. All the real numbers you’re used to working with are already complex numbers — they simply have zero imaginary part.

• To add or subtract complex numbers, add the corresponding real and imaginary parts. For example, 2 + 3i plus 4 + 5i is 6 + 8i.

• To multiply two complex numbers a + bi and c + di, just FOIL out the product (a + bi)(c + di) and use the fact that i^{2} = −1. Then collect like terms.

• The familiar exponential function f(x) = e^{x} takes real-valued input. However, it can be extended to take complex-valued input. All the usual rules for exponents apply, so

e^{a+bi} = e^{a}e^{bi}.

We compute e^{a} as always — this is the same exponential function as always. The question is, what does it mean to raise e to an imaginary power? I assert to you that we write

e^{bi} = cos(b) + i sin(b)

where the cosine and sine functions are as usual. This famous formula is called Euler’s formula (Euler is pronounced Oiler). You can read all about this formula on Wikipedia — also see their nice article on the complex numbers.

Given these facts, we can simply write down what e^{i(α+β)} is: the sum and difference formulas for sine and cosine fall out as a consequence. Using the usual rules for exponents, we can write this as

e^{i(α+β)} = e^{iα}e^{iβ}.

Now all we need to do is write out the two sides using Euler’s formula. The left-hand side is

e^{i(α+β)} = cos(α + β) + i sin(α + β).

Using the definition, FOILing, and collecting like terms, the right-hand side is

e^{iα}e^{iβ} = (cos α + i sin α)(cos β + i sin β)
= (cos α cos β − sin α sin β) + i(sin α cos β + cos α sin β).

Equating real and imaginary parts of the left-hand side and the right-hand side gives us, two for the price of one, the familiar sum identities for sine and cosine:

sin(α + β) = sin α cos β + cos α sin β
cos(α + β) = cos α cos β − sin α sin β.

Repeat this for e^{i(α−β)} to get the difference identities. You can do that — just remember that cosine and sine are even and odd functions, respectively, so cos(−β) = cos(β) and sin(−β) = − sin(β).

In summary, we have:

sin(α ± β) = sin α cos β ± cos α sin β
cos(α ± β) = cos α cos β ∓ sin α sin β.