Even Derivatives of Cotangent of Pi Z at One Fourth/Lemma

Even Derivatives of Cotangent of Pi Z at One Fourth: Lemma

Let $z \ne \paren {4 n + 1} \dfrac 1 4$

Then:

$\ds \map \tan {\pi z + \dfrac \pi 4} = \map \sec {2 \pi z} + \map \tan {2 \pi z}$

where:

$\sec$ and $\tan$ are secant and tangent respectively.

Proof

\(\ds \map \tan {\pi z + \dfrac \pi 4}\)

\(=\)

\(\ds \dfrac {\map \tan {\pi z } + \map \tan {\dfrac \pi 4 } } {1 - \map \tan {\pi z } \map \tan {\dfrac \pi 4 } }\)

Tangent of Sum

\(\ds \)

\(=\)

\(\ds \dfrac {1 + \map \tan {\pi z } } {1 - \map \tan {\pi z } }\)

Tangent of 45 Degrees

\(\ds \)

\(=\)

\(\ds \dfrac {1 + \map \tan {\pi z } } {1 - \map \tan {\pi z } } \times \dfrac {\map \cos {\pi z } } {\map \cos {\pi z } }\)

multiplying top and bottom by $\map \cos {\pi z }$

\(\ds \)

\(=\)

\(\ds \dfrac {\map \cos {\pi z } + \map \sin {\pi z } } {\map \cos {\pi z } - \map \sin {\pi z } }\)

Tangent is Sine divided by Cosine

\(\ds \)

\(=\)

\(\ds \dfrac {\map \cos {\pi z } + \map \sin {\pi z } } {\map \cos {\pi z } - \map \sin {\pi z } } \times \dfrac {\paren {\map \cos {\pi z } + \map \sin {\pi z } } } {\paren {\map \cos {\pi z } + \map \sin {\pi z } } }\)

multiplying top and bottom by $\paren {\map \cos {\pi z } + \map \sin {\pi z } }$

\(\ds \)

\(=\)

\(\ds \dfrac {\map {\cos^2} {\pi z } + 2 \map \sin {\pi z } \map \cos {\pi z } + \map {\sin^2} {\pi z } } {\map {\cos^2} {\pi z } - \map {\sin^2} {\pi z } }\)

\(\ds \)

\(=\)

\(\ds \dfrac {1 + \map \sin {2 \pi z } } {\map \cos {2 \pi z } }\)

Sum of Squares of Sine and Cosine, Double Angle Formula for Sine and Double Angle Formula for Cosine

\(\ds \)

\(=\)

\(\ds \map \sec {2 \pi z} + \map \tan {2 \pi z}\)

Definition of Secant Function and Definition of Tangent Function

$\blacksquare$