Sum of Sines and Cosines — Beautiful Closed-Form Formulas

Introduction

The sums

and

have surprisingly elegant closed-form expressions. These identities are extremely useful in A-level/Further Maths trigonometry, complex numbers, and Fourier-type expansions.

Below are the two main formulas, followed by two different proofs:

1. a classical trigonometric proof using sum-to-product and telescoping cancellation

2. a short, elegant proof using geometric series and complex numbers.

1. Theorem — Sum of Sines

Proof (Trigonometric Method)

The result is very reminiscent of the sum of an arithmetic series formula,

1+2+3+···+n = n(n+1)/2. We begin by writing out the sum forwards and in reverse, let

Adding, we get

On each [...] using the sum-to-product formula,

we get,

Factorising,

Now comes the key idea:

Multiply both sides by sin⁡(x/2)

In the [...] each sin(*)cos(*) term can be rewritten using the product-to-sum formula,

So we get,

Now we note that sin((2x-nx)/2) = -sin((-2x+nx)/2) and sin((4x-nx)/2) = -sin((-4x+nx)/2) and so forth. This means that all terms inside [...] cancel out except the first and last (telescopic cancelling). So we get,

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2. Theorem — Sum of Cosines

Proof (Trigonometric Method)

Following exactly the same forward-and-reverse strategy, we write

Adding, we get

Applying the sum-to-product formula,

we get,

Factorising,

Again, multiplying by sin⁡(x/2) and using telescoping cancellation leads to:

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3. A Beautiful Result

If we now divide the two formulas, we get the elegant result

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4. Alternative Proof Using Complex Numbers

Both formulas can also be proved by taking the real and imaginary parts of the geometric series,

From the summation formula this becomes,

Therefore,

The imaginary part of the RHS is:

Hence,

In equation * if we take the real part, we get

Hence,

Conclusion

These closed-form expressions for the sums of sines and cosines are not only elegant, but also extremely powerful. They can be derived using classical trigonometry or, much more compactly, via the geometric series in complex form.

This topic is part of a wider set of techniques involving periodic sequences, complex numbers and Fourier-type expansions—useful for A-level Maths, Further Maths, STEP, and undergraduate study.