Question

if a=cos2x + sin2x and b=cos2y + isin2y.then find (a + b)

Answer 1

I have a sneaky suspicion that your first equation is
a = cos 2x + i sin 2x

a+b = (cos 2x + cos 2y) + i(sin 2x + sin 2y)

recall cosA + cosB = 2cos( (A+B)/2 )cos( (A-B)/2)
so cos 2x + cos 2y = 2 cos(x+y) cos(x-y)
recall sinA + sinB = 2sin( (A+B)/2)cos( (A+B)/2)
so sin 2x + sin 2y = 2sin(x+y)cos(x+y)

then a+b = (cos 2x + cos 2y) + i(sin 2x + sin 2y)

= 2 cos(x+y) cos(x-y) + 2i sin(x+y)cos(x+y)