PROVE sin^-1(3/5)+sin^-1(5/13)=sin^-1(56/65)?

x = arcsin(3/5)
y = arcsin(5/13)
sin x = 3/5, cos x = 4/5
sin y = 5/13, cos y = 12/13
sin(x+y) = (3/5)(12/13)+(4/5)(5/13)
= 36/65 + 20/65
= 56/65
x+y = arcsin(56/65)

　
sin(sin⁻¹(3/5) + sin⁻¹(5/13))
= sin(sin⁻¹(3/5)) cos(sin⁻¹(5/13)) + cos(sin⁻¹(3/5)) sin(sin⁻¹(5/13))
= (3/5) √(1−25/169) + √(1−9/25) (5/13)
= (3/5) (12/13) + (4/5) (5/13)
= 36/65 + 20/65
= 56/65
= sin(sin⁻¹(56/65))

sin(sin⁻¹(3/5) + sin⁻¹(5/13)) = sin(sin⁻¹(56/65))
⇒ sin⁻¹(3/5) + sin⁻¹(5/13) = sin⁻¹(56/65)