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

TL;DR

This paper derives multiple formulas for the k'th derivatives of powers of trigonometric and hyperbolic functions, using complex definitions and polynomial expressions involving only basic functions.

Contribution

It introduces new polynomial-based expressions for derivatives that avoid complex arguments, simplifying calculations for even and odd derivatives.

Findings

Derived formulas using complex definitions and binomial theorem.

Presented polynomial-based expressions involving only basic functions.

Provided explicit formulas for even and odd derivatives.

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.