Решение

Since waves are superimposed,

$$E_R = E_{w1} + E_{w2}$$

$$E_R = E_1~\sin{[\omega (t-z/c)+\varphi_1]} + E_2~\sin{[\omega (t-z/c)+\varphi_2]}$$

$$E_R = (E_1\cos{\varphi_1}+E_2\cos{\varphi_2})\sin{[\omega (t-z/c)]} + (E_1\sin{\varphi_1}+E_2\sin{\varphi_2})\cos{[\omega (t-z/c)]}$$

Let's suppose that

$$E_1\cos{\varphi_1}+E_2\cos{\varphi_2} = E \cos{\varphi} = E_x$$

and

$$E_1\sin{\varphi_1}+E_2\sin{\varphi_2} = E \sin{\varphi} = E_y$$

and considering the trigonometric identity $\sin{(x+y)} = \sin{x}\cos{y} + \cos{x}\sin{y}$,

$$E_R = E \sin{[\omega (t-z/c)+\varphi]}$$

As $E = \sqrt{{E_x}^2 + {E_y}^2}$ and taking in account that $\cos{(x-y)} = \cos{x}\cos{y} + \sin{x}\sin{y}$