In a `Delta ABC`, if `tan.(A)/(2)=(5)/(6), tan.(B)/(2)=(20)/(37)`, then which of the following is/are correct ?

Explanation

`tanC/2=tan{(180^(@)-(A+B))/(2)}`
`=tan{90^(@)-(A/2+B/2)}`
`=cot(A/2+B/2)=1/(tan(A/2+B/2))`
`=(1-tanA/2tanB/2)/(tanA/2+tanB/2)=(1-5/6 xx 20/37)/(5/6 + 20/37)`
`=(6 xx 37 -5 xx 20)/(5 xx 37+6 xx 20)=(222-100)/(185+120)`
`=122/305=2/5`
Now `sinA=(2tanA/2)/(1+tan^(2)A/2)=(2 xx 5/6)/(1+(5/6)^(2))`
`=5/3 xx 36/61 = 60/61`
`sinB=(2tanB/2)/(1+tan^(2)B/2)=(2 xx 20/37)/(1+(20/37)^(2))`
`=40/37 xx 1369/1769=1480/1769`
and `sinC=(2tanC/2)/(1+tan^(2)C/2)=(2 xx 2/5)/(1+(2/5)^(2))`
`=4/5 xx 25/29 = 20/29`
`therefore a:b:c=sinA:sinB:sinC`
`=60/61:1480/1769:20/29`
`=60 xx 29 : 1480 : 20 xx 61`
`=1740 : 1480:20 xx 61`
`=1740 : 1480:20 xx 61`
`=1740:1480:1220=87:74:61`
Let a=87k, b=74k, c=61k
Now `b-a=74k-87k=-13k`
`therefore b-a=c-b`
`therefore a,b,c` are in A.P.