correct me if im wrong but shouldnt the right side be multiplied by 2??

Then your textbook is wrong. sin((x+y)/2)cos((x-y)/2) certainly doesn't equal sin x + sin y. It equals (sin x + sin y)/2.

Let's assume the question asks to prove sin x + sin y = 2sin((x+y)/2)cos((x-y)/2). There is nothing esoteric about this proof. Just use the fact that sin(A+B)=sin A cos B + cos A sin B and cos(A-B)=cos A cos B + sin A sin B.

it's an elementary proof if you rewrite sine as an exponential using euler's formula

I'm currently in MCR3U(E) [this is an Enriched course that is offered in my school], we basically do MCR3U1 + MHF4U1 (No Calculus), I'd say that trigonometric identities are the hardest thing in the entire course (especially with tangents, which have some weird identities that are similar, but not so similar in some other aspects, to the sine and cosint identities).

Yes, if anyone wishes to go into some form of Math major, you must learn how to do proofs, properly. These are the basics:

sin(A+B)=sinAcosB+sinBcosA
sin(A-B)=sinAcosB-sinBcosA
cos(A+B)=cosAcosB-sinAsinB
cos(A-B)=cosAcosB+sinAsinB
sin(2A)=2sinAcosA
cos(2A)=(cosA)^2-(sinA)^2
sinA/cosA=tanA
(sinA)^2+(cosA)^2=1
(tanA)^2+1=(secA)^2

The identity "e^i*pi + 1 = 0" is something I do not understand, although we do not care about e at this moment in my MCR3UE course