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if 2tanA=3tanB,prove that tan(A-B)=sin2B/5-cos2B
Hello student,Given that 2 tan A = 3 tan B,So, tan A = 3/2 tan B ….. (i)
Dear Student,Please find below the solution to your problem.If 2tanA=3tanB or tanA=(3/2).tanBProve that tan(A-B)=sin2B/(5-cos2B)LHS.tan(A-B)=(tanA-tanB)/(1+tanA.tanB), put tanA=(3/2).tanB.=[(3/2)tanB-tanB]/[1+(3/2)tan^2B]=[(1/2).tanB]/[(2cos^2B+3sin^2B)/2cos^2B].=[(sinB/2cosB)×2cos^2B]/[2cos^2B+3sin^2B].=[sinBcosB]/(2cos^2B+3sin^2B)=(sin2B)/2.(2cos^2B+3sin^2B=(sin2B)/(4cos^2B+6sin^2B)=(sin2B)/[4(1-sin^2B)+6sin^2B]=(sin2B)/[4–4sin^2B+6sin^2B]=(sin2B)/[4+2sin^2B) , put 2sin^2B=1-cos2B=(sin2B)/[4+1-cos2B]=(sin2B)/(5-cos2B) , Proved.Thanks and Regards
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