Trigonometric question

If sin alpha+sin beta=a and cos alpha+cos beta =b , show that
1. sin(alpha+beta)=2ab/a^2+b^2

a^2 means 'a square'

sin α + sin ß = a (i)
cos α + cos ß = b (ii)

(i) 2sin (α+ß)/2 cos (α-ß)/2 = a

(ii) 2cos (α+ß)/2 cos (α-ß)/2 = b → cos (α-ß)/2 = b / 2cos (α+ß)/2

Substitute in (i) :

2b sin (α+ß)/2 / 2cos (α+ß)/2 = a → b tan (α+ß)/2 = a →

tan (α+ß)/2 = a/b

Now use the formula sin 2t = 2tan t / ( 1 + tan²t ) , with t = (α+ß)/2 :

sin (α+ß) = (2a/b) / ( 1 + a²/b² ) = 2ab² / b ( b² + a² ) = 2ab / ( a² + b² )