Question

What is tan(x)-5=0 (With steps)

Answer 1

$\tan x-5=0$
$\tan x=5$

Now before taking the inverse tangent of both sides, recall that

$\tan x=\tan(x\pm\pi)=\tan(x\pm2\pi)=\cdots=\tan(x\pm n\pi)$

where $n$ is any integer. This means that when taking the inverse tangent of $\tan x$, instead of just $x$, we would end up with

$\arctan(\tan x)=\arctan(\tan(x+n\pi))=x\pm n\pi$

So after doing so in the equation above, we're left with

$x\pm n\pi=\arctan5\implies x=\arctan5\pm n\pi$

As $n$ can be any integer, including negative integers, we can simply write

$x=\arctan5+n\pi,\,n\in\mathbb Z$