Mathematical Identities – Elite Math Program

Consider the innocent-looking sum

S = cos x + cos 2 x + cos 3 x + \dots + cos n x .

At first glance, there is nothing geometric about it. The terms are cosines, not powers of a common ratio.

But there is a geometric series hiding inside.

The Key Idea

Euler’s formula says

e^{i x} = cos x + i sin x .

Therefore, the real part of $e^{i k x}$ is $cos k x$ . Hence

S = Re (e^{i x} + e^{2 i x} + e^{3 i x} + \dots + e^{n i x}) .

Now look carefully at the expression inside the parentheses. It is a geometric series!

Its first term is $e^{i x}$ , and its common ratio is also $e^{i x}$ .

Sum the Geometric Series

Using the finite geometric-series formula,

e^{i x} + e^{2 i x} + \dots + e^{n i x} = e^{i x} \frac{1 - e^{n i x}}{1 - e^{i x}} .

This already proves that our trigonometric sum comes from a geometric series. But we can simplify it further.

A Useful Identity

For any real number $t$ ,

1 - e^{i t} = - 2 i e^{i t / 2} sin (\frac{t}{2}) .

Apply this identity to both the numerator and denominator. After cancellation, we obtain

e^{i x} + e^{2 i x} + \dots + e^{n i x} = \frac{sin (\frac{n x}{2})}{sin (\frac{x}{2})} e^{i \frac{(n + 1) x}{2}} .

Now take the real part. Since the real part of $e^{i θ}$ is $cos θ$ , we arrive at

cos x + cos 2 x + \dots + cos n x = \frac{sin (\frac{n x}{2}) cos (\frac{(n + 1) x}{2})}{sin (\frac{x}{2})} .

So the final formula is

cos x + cos 2 x + \dots + cos n x = \frac{sin (\frac{n x}{2}) cos (\frac{(n + 1) x}{2})}{sin (\frac{x}{2})} .

This formula applies whenever $sin (x / 2) \neq 0$ . If $x$ is a multiple of $2 π$ , every cosine equals 1, so the original sum is simply $n$ .

There Is Geometry Behind the Geometric Series

The connection is even more interesting than the algebra suggests.

Each complex number

e^{i k x} = cos k x + i sin k x

can be viewed as a vector of length 1 making an angle $k x$ with the positive horizontal axis.

Thus the vectors

e^{i x}, e^{2 i x}, e^{3 i x}, \dots, e^{n i x}

all have the same length, and each successive vector is obtained by rotating the previous one through exactly the same angle $x$ .

Place these vectors head-to-tail. They form a turning polygonal chain. Their vector sum is

e^{i x} + e^{2 i x} + \dots + e^{n i x} .

And what is the horizontal component of this vector?

Exactly

cos x + cos 2 x + \dots + cos n x .

So our sum of cosines really does have a geometric meaning.

And the Sines Come for Free

The imaginary part of exactly the same geometric series gives another classical identity:

sin x + sin 2 x + \dots + sin n x = \frac{sin (\frac{n x}{2}) sin (\frac{(n + 1) x}{2})}{sin (\frac{x}{2})} .

One geometric series has given us two trigonometric identities.

The Takeaway

A sum such as

cos x + cos 2 x + \dots + cos n x

does not look remotely like a geometric series.

But complex numbers reveal what is hidden:

trigonometric sum \to complex exponentials \to geometric series \to closed formula.

Sometimes the hardest part of a problem is not doing the calculation. It is recognizing what the calculation really is.

Another Unexpected Side of Trigonometry

Writing a sum of cosines as part of a geometric series reveals algebra hidden inside trigonometry. There is another beautiful example of this idea in the historical problem of actually computing sines and cosines.

Long before electronic calculators, Newton developed infinite series that turned $\sin (x)$ and $\cos (x)$ into expressions that could be evaluated using arithmetic.

Continue exploring: How Newton Computed Sines and Cosines Without a Calculator

Tag: Mathematical Identities

A Sum of Cosines Is a Geometric Series—Could You Believe It?