# A tangent identity

We all know that the Chebyshev polynomials of the first kind can be defined as

When I was fiddling about with identities associated with Machin’s formula I came across an identity for tangents, which I’m sure is well known, but which I’d never seen before. Here it is:

This can be more precisely written as follows:

and

And these two identities can be shoehorned into one ugly (but general!) expression:

The identity is not hard to prove by induction. As is already used for the Chebyshev polynomials, we shall write for and then write

(where stands for numerator and denominator respectively.) Then by the addition formula for , we have

.

Writing on the right in terms of and , and then multiplying through by produces

.

Recall that

and

Now consider the the numerator of :

.

The coefficient of will be

.

The denominator of is

and the coefficient of will be

.

Note that the constant coefficient is

These results, plus the trivial result for , prove the identity.