I was reading an article [1] that refers to “a well-known trigonometric series” that I’d never seen before. This paper cites [2] which gives the series as

Note that the right hand side is not a series in φ but rather in sin φ.

## Motivation

Why might you know sin φ and want to calculate sin *m*φ / cos φ? This doesn’t seem like a sufficiently common task for the series to be well-known. The references are over a century old, and maybe the series were useful in hand calculations in a way that isn’t necessary anymore.

However, [1] was using the series for a theoretical derivation, not for calculation; the author was doing some hand-wavy derivation, sticking the difference operator *E* into a series as if it were a number, a technique known as “umbral calculus.” The name comes from the Latin word *umbra* for shadow. The name referred to the “shadowy” nature of the technique which wasn’t make rigorous until much later.

## Convergence

The series above terminates if *m* is an even integer. But there are no restrictions on *m*, and in general the series is infinite.

The series obviously has trouble if cos φ = 0, i.e. when φ = ±π/2, but it converges for all *m* if −π/2 < φ < π/2.

## Tangent

If *m* = 1, sin *m*φ / cos φ is simply tan φ. The function tan φ has a complicated power series in φ involving Bernoulli numbers, but it has a simpler power series in sin φ.

## References

[1] G. J. Lidstone. Notes on Everett’s Interpolation Formula. 1922

[2] E. W. Hobson. A Treatise on Plane Trigonometry. Fourth Edition, 1918. Page 276.