[tex]\theta[/tex] often denotes an angle, where [tex]0\le\theta<2\pi[/tex] is conventionally used.
Adding [tex]2k\pi[/tex], where [tex]k[/tex] is any integer, is the same as saying that [tex]\theta[/tex] is equivalent to [tex]\theta\pm2\pi,\theta\pm4\pi,\theta\pm6\pi[/tex] etc.
This is the case because a complete revolution has an angle measure of [tex]2\pi[/tex], which means that adding any multiple of [tex]2\pi[/tex] to a given angle results in the same angle.
Example:
If [tex]\theta=\pi[/tex], then [tex]\cos\pi=-1[/tex]. This is also true if [tex]\theta=\pi+2\pi=3\pi[/tex], which gives [tex]\cos3\pi=\cos(\pi+2\pi)=\cos\pi\cos2\pi-\sin\pi\sin2\pi=(-1)(1)-(0)(0)=-1[/tex].
