Template:EqualOrNegativeIdenticalTrigonometricFunctionsSolutions

The table below shows how two angles $\theta$ and $\varphi$ must be related if their values under a given trigonometric function are equal or negatives of each other.

Equation			if and only if	Solution							where...	Also a solution to
$\sin \theta$	$=$	$\sin \varphi$	$\iff$	$\theta =$	$(-1)^{k}$	$\varphi$	$+$		$\pi k$		for some $k\in \mathbb {Z}$	$\csc \theta =\csc \varphi$
$\cos \theta$	$=$	$\cos \varphi$	$\iff$	$\theta =$	$\pm \,$	$\varphi$	$+$	$2$	$\pi k$		for some $k\in \mathbb {Z}$	$\sec \theta =\sec \varphi$
$\tan \theta$	$=$	$\tan \varphi$	$\iff$	$\theta =$		$\varphi$	$+$		$\pi k$		for some $k\in \mathbb {Z}$	$\cot \theta =\cot \varphi$
$-\sin \theta$	$=$	$\sin \varphi$	$\iff$	$\theta =$	$(-1)^{k+1}$	$\varphi$	$+$		$\pi k$		for some $k\in \mathbb {Z}$	$-\csc \theta =\csc \varphi$
$-\cos \theta$	$=$	$\cos \varphi$	$\iff$	$\theta =$	$\pm \,$	$\varphi$	$+$	$2$	$\pi k$	$+\,\;\pi$	for some $k\in \mathbb {Z}$	$-\sec \theta =\sec \varphi$
$-\tan \theta$	$=$	$\tan \varphi$	$\iff$	$\theta =$	$-$	$\varphi$	$+$		$\pi k$		for some $k\in \mathbb {Z}$	$-\cot \theta =\cot \varphi$
$|\sin \theta |$	$=$	$|\sin \varphi |$	$\iff$	$\theta =$	$\pm$	$\varphi$	$+$		$\pi k$		for some $k\in \mathbb {Z}$	$|\tan \theta |=|\tan \varphi |$
	⇕											$|\csc \theta |=|\csc \varphi |$
$|\cos \theta |$	$=$	$|\cos \varphi |$										$|\sec \theta |=|\sec \varphi |$
												$|\cot \theta |=|\cot \varphi |$