# Math Help - Equation

1. ## Equation

Hi,

I must show that $(\forall k \in \mathbb{Z}) \forall x \in ]-\frac{\pi}{2}+k\pi;\frac{\pi}{2}+k\pi[: Arctan(tanx)=x-k\pi.$

I know just that $Arctan(tanx)=x$ when $x \in ]-\frac{\pi}{2}+k\pi;\frac{\pi}{2}+k\pi[$

Can you help me???

2. Originally Posted by lehder
Hi,

I must show that $(\forall k \in \mathbb{Z}) \forall x \in ]-\frac{\pi}{2}+k\pi;\frac{\pi}{2}+k\pi[: Arctan(tanx)=x-k\pi.$

I know just that $Arctan(tanx)=x$ when $x \in ]-\frac{\pi}{2}{\color{red}+k\pi};\frac{\pi}{2}{\col or{red}+k\pi}[$
No, I think that's wrong (and contradicts what you are required to prove). Instead, you know that $Arctan(tanx)=x$ when $x \in ]-\frac{\pi}{2};\frac{\pi}{2}[$

And then you know something more: you know that $\tan$ has period $\pi$, therefore you can add / subtract $\pi$ an integral number of times until the value of an $x\in]-\frac{\pi}{2}+k\pi;\frac{\pi}{2}+k\pi[$ actually lies in $]-\frac{\pi}{2};+\frac{\pi}{2}[$.