tana=2tanB show that sin(a +b)/sin(a-b)=3

Question

given tana=2tab

Answer 1

sin(A+B)/sin(A-B)

(sinA.cosB + cosA.sinB)/(sinA.cosB - cosA.sinB)

now divide above and below by cosA.sinB,

(tanA.cotB + 1)/(tanA.cotB - 1)

now substitute for tanA = 2tanB,

(2tanB.cotB + 1)/(2tanB.cotB - 1)

(2 + 1)/(2 - 1)    (using cotB = 1/tanB)

=3

Answer 2

We have given cotB-cotA=y.....1 TanA-tanB=x 1/cotA-1/cotB=x CotB-cotA/cotAcotB=x From eq.1 Y/cotAcotB=x CotAcotB=y/x........2 So Show that cot(A-B)=x+y/xy CotAcotB+1/cotB-cotA Y/x+1/y Y+x/xy hence proved