Given that Sin(A+B)=2Cos(A-B) and tan A = 1/3 ,

Question

Find the exact value of tan B.

Answer 1

if tan A = 1/3, then by Pythagoras,
sin A = 1/√10
cos A = 3/√10

if sin(A+B) = 2cos(A-B)
sinAcosB + cosAsinB = 2(cosAcosB + sinAsinB)
(1/√10)cosB + (3/√10)sinB = (6/√10)cosB + (2/√10)sinB
multiply by √10 and simplify
cosB + 3sinB = 6cosB + 2sinB
sinB = 5cosB
sinB/cosB = 5/1
tanB = 5/1

then cosB = 1/√26