Tangent Function is Periodic on Reals

Theorem

The real tangent function is periodic with period $\pi$.

This can be written:

$\tan x = \map \tan {x \bmod \pi}$

where $x \bmod \pi$ denotes the modulo operation.

$\ds \map \tan {x + \pi}$

$=$

$\ds \frac {\map \sin {x + \pi} } {\map \cos {x + \pi} }$

Definition of Real Tangent Function

$\ds $

$=$

$\ds \frac {-\sin x} {-\cos x}$

$\ds $

$=$

$\ds \tan x$

$\map {D_x} {\tan x} = \dfrac 1 {\cos^2 x}$

provided $\cos x \ne 0$.

From Shape of Cosine Function, we have that $\cos > 0$ on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

From Derivative of Monotone Function, $\tan x$ is strictly increasing on that interval, and hence can not have a period of less than $\pi$.

Hence the result.

$\blacksquare$

1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 16.5 \ (2)$
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Shifts and periodicity
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Shifts and periodicity