What is the meaning of Trigonometry Identities being true for all values?

$$\tan\theta=1$$ 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.


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 $$\arcsin(\sin z)=z\qquad(z\in\Omega)\ .$$

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Trigonometry

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