Transverse nature of EM waves

As E (x.t) not of y and z

E_xa – E_x’a = 0

(E_x – E_x’)a = 0

Lither a = 0 or not possible

E_x – E_x’ = 0 possible

-->  E_x = E_x’ not possible as E(x, t)

Or E_x = 0 and E_x’ = 0 possible

It shows that at no point electric field has component along x-direction. In other words at all the points electric field in perpendicular to x-axis.

Similarly using equation and the condition B(x, t) use can prove that magnetic field is also perpendicular to x-axis hence EM wave is transverse in nature.