On the representation of tan(nθ) by powers of tanθ

ABSTRACT

Contrary to the assertion in a recent paper that the formula for the tangent function of multiple angles is not widely known, whereas the corresponding ones for the sine and the cosine functions are common knowledge, it is pointed out that an entirely equivalent expression to one given in this paper may be found in a standard treatise on elementary trigonometry first published in the late nineteenth century. Comparison of the two expressions immediately shows two recommended changes in the recent formula as stated. This assertion may be checked against simple special cases either derived, or set as student exercises, in the sections on multiple and submultiple angles in three text-books widely used in schools and universities in the early twentieth century. An expression of a more general type, for tan(θ ₁+ θ ₂+ … + θ _n ), which is probably of wider practical use, is derived by induction in one of these texts. The derivation of an equivalent expression in terms of a terminating continued fraction perhaps presents an interesting challenge.