Consider two light waves moving in z direction represented by two mutually perpendicular electric field vectors (with non-zero amplitudes):${\mathrm{E}}_{\mathrm{x}}={\mathrm{A}}_{\mathrm{x}}\cos \left({\mathrm{\omega }}_{\mathrm{x}}\mathrm{t}+{\mathrm{\varphi }}_{\mathrm{x}}\right)$ and ${\mathrm{E}}_{\mathrm{y}}={\mathrm{A}}_{\mathrm{y}}\sin \left({\mathrm{\omega }}_{\mathrm{y}}\mathrm{t}+{\mathrm{\varphi }}_{\mathrm{y}}\right)$. Their superposition will result in a plane polarized light, if 

Ratio of $\cos \left({\mathrm{\omega }}_{\mathrm{x}}\mathrm{t}+{\mathrm{\varphi }}_{\mathrm{x}}\right)\text{ and }\sin \left({\mathrm{\omega }}_{\mathrm{y}}\mathrm{t}+{\mathrm{\varphi }}_{\mathrm{y}}\right)$ should be non-zero finite constant.

The two perpendicular vibrating light waves form a plane polarized light if their frequencies are equal ${\mathrm{\omega }}_{\mathrm{x}}={\mathrm{\omega }}_{\mathrm{y}}$ and have a phase difference of zero. The amplitude of the waves may or may not be equal.

${\mathrm{E}}_{\mathrm{x}}={\mathrm{A}}_{\mathrm{x}}\cos \left({\mathrm{\omega }}_{\mathrm{x}}\mathrm{t}+{\mathrm{\varphi }}_{\mathrm{x}}\right)={\mathrm{A}}_{\mathrm{x}}\sin \left({\mathrm{\omega }}_{\mathrm{x}}\mathrm{t}+{\mathrm{\varphi }}_{\mathrm{x}}+\frac{\mathrm{\pi }}{2}\right)$

and 

${\mathrm{E}}_{\mathrm{y}}={\mathrm{A}}_{\mathrm{y}}\sin \left({\mathrm{\omega }}_{\mathrm{y}}\mathrm{t}+{\mathrm{\varphi }}_{\mathrm{y}}\right)$

Since, ${\mathrm{\omega }}_{\mathrm{x}}={\mathrm{\omega }}_{\mathrm{y}}$ , the phase difference should be zero.

${\varphi }_x+\frac{\pi }{2}-{\varphi }_{\mathrm{y}}=0$

The values which satisfy above equation are ${\mathrm{\varphi }}_{\mathrm{x}}=\frac{\mathrm{\pi }}{2},{\mathrm{\varphi }}_{\mathrm{y}}=\mathrm{\pi }$