Relating the diffraction angle to the angle from the “zeroth order beam”

What [![enter image description here](https://i.stack.imgur.com/476kC.jpg)](https://i.stack.imgur.com/476kC.jpg)I’ve been asked relates to diffraction from scattering atoms in materials with periodic atom array structures.


“For two scattered in a beam of radiation that are a distance d apart, show that the angle of a constructively diffracted beam away from the straight through beam is given by sin(theta)=n(lambda)/d.”


Obviously This is not the same angle used in the derivation of braggs law, however in my working (attached) I thought I could use it to relate the 2 angles via trigonometry. I’ve tried using trig identities like sin2x=2sinxcosx as well as the sin rule with Pythagoras but I, just getting no where. Could anyone possibly give any insight as to what I’m doing wrong/ misunderstanding here?


This is a non conventional derivation (not just a Bragg’s law derivation) but is still relevant (I would have thought this would have been a useful relation to know). I just can’t seem to see how Angles psi and theta relate. I would greatly appreciate any ideas which might help me figure out what I’m doing wrong here. [![enter image description here](https://i.stack.imgur.com/hGyGg.jpg)](https://i.stack.imgur.com/hGyGg.jpg)


