Adding, the Cosine

Adding the two equations we have

$$e^{\imath \theta} + e^{-\imath \theta} = \cos \theta - \imath \sin \theta + \cos \theta + \imath \sin \theta = 2 \cos \theta$$

so

$$\cos \theta = \frac{ e^{\imath \theta} + e^{-\imath \theta} }{2}$$ (4)