Explicit formulas related to Euler's product expansion for cosine function

TL;DR

This paper derives explicit formulas and identities related to Euler's product expansion for the cosine function using elementary methods, including continued fractions and series expansions.

Contribution

It introduces new elementary derivations of identities involving derivatives of tanx, log coshx, and a formula for π^2 from series expansions of tan x.

Findings

Derived continued fractions for tan x

Identities involving derivatives of tan x

Expressions for log cosh x

Abstract

In this paper, we derive by using elementary methods some continued fractions, certain identities involving derivatives of tanx, several expressions for log coshx and an identity for {\pi}2, from a series expansion of tan x, which gives the product expansion of the cosine function.