If a= 2cosx+ isinx and b = 2cosy +isiny then prove that a+b = 2cos(x-y)(cos(x+y)+isin(x+y))

Answer 1

just plug and chug

a+b = 2cosx+i sinx + 2cosy + i siny
= 2(cosx+cosy) + (sinx+siny)i
Now,
(sinx+cosx)^2 = sin^2x+2sinx cosx+cos^2x
= 1+2sinx cosx
similarly, (siny+cosy)^2 = 1+2siny cosy

Since 2^2-b^2 = (a+b)(a-b), we have

(sinx+cosx)^2 - (siny+cosy)^2 = ((sinx+cosx)+(siny+cosy))((sinx+cosx)(siny+cosy))

Try expanding that out, and use your sum/difference formulas to get to the right side.