We will use Faraday's law :
nabla times vec{E} + {partial \vec{B}}/{partial t} = 0
to determine \vec{B}. We first calculate the curl of \vec{E} :
nabla times vec{A} = | (hat i, hat j, hat k),(partial/{partial x}, partial/{partial y}, partial/{partial z}),(2E_0 cos kz cos omega t,0,0)|
= hat j partial/{partial z }(2E_0 cos kz cos omega t) = -2E_0 k sin kz cos omega t hat{k}
Thus
{partial vec{B}}/{partial t} = 2E_0 k sin kz cos omega t hat{k}
Integration yields
vec B = 2{E_0 k}/omega sin kz sin omega t hat{k} = 2{E_0 }/c sin kz sin omega t hat{k}