I wasn’t sure whether to post this issue in a physics forum or here in this maths forum, because although it relates to physics is appears to be grounded in maths, i.e. Euler theorem. Therefore, I was wondering if anybody could help me resolve some issues with the following derivation, taken from a textbook, for 2 waves in superposition. The derivation is linked to the Young’s double-slit experiment, which acts as a precursor to quantum theory

[1] [tex]\Psi = A exp \; i \left[ k \left( r- \frac{d}{2}sin \theta\right) - wt \right] + A exp \; i \left[ k \left( r+ \frac{d}{2}sin \theta\right) - wt \right] [/tex]

[2] [tex]\Psi = A exp \; i \left( kr-wt\right) \left( exp \left[ ik \left( \frac{d}{2} \right)sin \theta \right] + exp \left[ -ik \left( \frac{d}{2} \right) sin \theta \right] \right)[/tex]

[3] [tex] \Psi = 2A exp \; i \left( kr-wt\right) cos \left[ k \left( \frac{d}{2} \right)sin \theta \right] [/tex]

Purely, by way of background, equation [1] represents 2 waves in superposition, hence the 2 parts, separated by the distance between the 2 slits [d]. The distance [r] is the mean distance to the screen where the interference pattern is seen. However, my first maths question relates to the premise of equation [1] being based on the following relationship and whether equation [1] has to consider both the real and imaginary components of the wave function?

*(I accept this might not be a pure maths issue)*

[4] [tex] e^{i \theta} = cos \theta + i.sin \theta [/tex]

While I think I follow the basic steps from [1] to [3], there appears to be an assumption when going from [2] to [3] that:

[5] [tex] \left( exp \left[ ik \left( \frac{d}{2} \right)sin \theta \right] + exp \left[ -ik \left( \frac{d}{2} \right) sin \theta \right] \right) = 2 cos \left[ k \left( \frac{d}{2} \right)sin \theta \right] [/tex]

So my second question is whether this based on the mathematical relationship?

[6] [tex] \Re (e^{i \theta}) = cos \theta = (e^{i \theta}+ e^{-i \theta})/2 [/tex]

In part, this goes back to the scope of whether equation [1] is intended to include the real and imaginary parts of the wave function. Would appreciate any knowledgeable insights to the maths and/or physics. Thanks