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`