On the derivatives of the powers of trigonometric and hyperbolic sine and cosine

Abstract

This work contains different expressions for the k'th derivative of the n'th power of the trigonometric and hyperbolic sine and cosine. The first set of expressions follow from the complex definitions of the trigonometric and hyperbolic sine and cosine, and the binomial theorem. The other expressions are polynomial-based. They are perhaps less obvious, and use only polynomials in sin(x) and cos(x), or in sinh(x) and cosh(x). No sines or cosines of arguments other than x appear in these polynomial-based expressions. The final expressions are dependent only on sin(x), cos(x), sinh(x), or cosh(x) respectively when k is even; and they only have a single additional factor cos(x), sin(x), cosh(x), or sinh(x) respectively when k is odd.