sin (arctan 1/2 + arctan 1/3)

sin (arctan 1/2) cos (arctan 1/3) + cos (arctan 1/2) sin (arctan 1/3)

arctan 1/2 is an angle who's tangent is 1/2. That angle would have a sin = 1/√5 and a cos= 2/√5.

arctan 1/2 is an agle who's tangent is 1/3. That angle would have sin = 1/√10 and a cos = 3/√10.

If these assumptions are all correct. Then I end up with

(1/√5)(3/√10) + (2/√5)(1/√10) = 5/√15

Please help me understand what I did wrong.