If `(sinA)/(sinB)=(sqrt(3))/2 and (cosA)/(cosB)=(sqrt(5))/2, 0 < A, B < pi/2,` then `tanA-tanB` is equal to

Explanation

Given `sinA/sinB=sqrt3/2" and " cosA/cosB=sqrt5/2,0ltA,Bltpi//2`
On dividing, we get
`tanA=sqrt3/sqrt5tanB …(i)`
Multiplying `(sinAcosA)/(sinBcosB)=sqrt15/4`
`rArr(tanA.sec^2B)/(tanA.sec^2A)=sqrt15/4`
or `sqrt3/sqrt5((1+tan^2B))/((1+tan^2A))=sqrt15/4" [using(i)]"`
or `4+4tan^2B=5+5tan^2A`
or `-1+4tan^2B=5xx3/5tan^2B`
or `tanB=pm1`
or `tanB=1 ( :'0ltBltpi/2)`
Now, `tanA+tanB =sqrt3/sqrt5+1=(sqrt3+sqrt5)/sqrt5`