What is the meaning of Trigonometry Identities being true for all values? "Trigonometric identities hold true for all the values of $\theta$". I can't understand this because there are some values which Trigonometric Identities are undefined. Given the Identities:
$$
\sec^2\theta-\tan^2\theta=1; |\sec\theta|\geq1
$$
$$
\forall\;\theta\in\mathbb{R}-\{(2n+1)\frac{\pi}{2}, n\in\mathbb{Z} \} 
$$
and
$$ 
\csc^2\theta-\cot^2\theta=1; |\csc\theta|\geq1
$$
$$
\forall\; \theta\in\mathbb{R}-     \{n\pi, n\in\mathbb{Z} \} $$ 
The first one, for example, $\tan\theta$ is undefined when $\cos\theta=0$. Probably there exists misunderstanding about concepts by me, feel free to correct. Perhaps, this was not the best example, if there were better ones, please answer me. But I would like to know if "Trigonometric identities hold true for all the values of $\theta$" is always true? 
 A: $$\tan\theta=1$$ is not an identity, it is an equation. Because it is true only for certain  values of $\theta$.
$$\tan\theta=\frac{\sin\theta}{\cos\theta}$$ is an identity because whenever the members are defined, they are equal.
This is what is meant by "holds true for all the values of $\theta$". 

If you prefer, there are no values of $\theta$ such that the two expressions differ.
A: This has to do with the principle of unique analytical continuation. In high school you prove from right triangles that for $0<x<{\pi\over2}$ you have $\cos^2 x+\sin^2 x=1$. Later you learn that the functions $x\mapsto\cos x$ and $x\mapsto\sin x$ can be extended to analytic functions on all of ${\mathbb C}$. The named principle then says that the identity
$$\cos^2 x+\sin^2 x=1\qquad\left(0<x<{\pi\over2}\right)$$
enforces
$$\cos^2z+\sin^2 z=1\qquad(z\in{\mathbb C})\ ,$$
and similarly for other such identities, as long as we can draw a connected region $\Omega\subset{\mathbb C}$ containing an arc or larger, where this identity is valid.
It is different with identities like
$$\arcsin(\sin x)=x\qquad\left(-{\pi\over2}\leq x\leq {\pi\over2}\right)\ .\tag{1}$$
Here $\arcsin$ is not a global inverse function of $\sin$, but is defined ad hoc by $(1)$, and is the inverse of $\sin$ on the interval $\bigl[-{\pi\over2},{\pi\over2}\bigr]$. One has to make detailed studies, where in ${\mathbb C}$ $\arcsin$ could be defined, and what a region $\Omega\subset{\mathbb C}$ could be such that
$$\arcsin(\sin z)=z\qquad(z\in\Omega)\ .$$
