Trigonometry

write tanB = sinB / cosB and cross multiply

sinB - n sin²AsinB = n sinA cosA cosB

sinB = nsinA ( sinAsinB+cosAcosB) = nsinA cos(A-B)

now subtract sinAcos(A-B) from both sides

R.H.S = (n-1)sinA cos(A-B)

L.H.S = sinB - sinAcos(A-B)=sinB - sinA (sinAsinB+cosAcosB)

= sinB - sin²A sinB - sinAcosAcosB

=sinB -(1-cos²A) sinB - sinAcosAcosB

=cos²AsinB - sinAcosAcosB

= cosA (cosAsinB - sinA cosB)

= - cosA sin(A-B)

so  - cosA sin(A-B) = (n-1)sinA cos(A-B)

tan(A-B) = (1-n) tanA