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Answer

(i)

sin(A-B)/cos(A+B) = (1/3)tan(A-B)

3sin(A-B)/cos(A+B) = sin(A-B)/cos(A-B)

3/cos(A+B) = 1/cos(A-B)

3cos(A-B) = cos(A+B)

3cosAcosB+3sinAsinB = cosAcosB-sinAsinB

2cosAcosB = -4sinAsinB

-1/2 = sinAsinB/cosAcosB

-1/2 = tanAtanB (proven)

 

(ii)

sinA = 1/2

A=30°,150°

tanA=0.57735,-0.57735

 

tanAtanB = -1/2

tanB = -1/(2tanA)

Thus, tanB=-0.86602,0.86602

B=40.893°,139.107°,220.893°,319.107°

cosB=0.75592,-0.75592

cosB=0.756,-0.756 (3 sig fig)

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