If `(sinx)/(siny)=1/2,(cosx)/(cosy)=3/2,` where `x,y in (0,pi/2),` then the value of `tan(x+y)` is equal to

Options

A. `sqrt(13)`
B. `sqrt(14)`
C. `sqrt(17)`
D. `sqrt(15)`

Answer

D

Explanation

`(sinx)/(siny)=(1)/(2),(cos x)/(cosy)=(3)/(2)`
`rArr (tan x)/(tany)=(1)/(3)`
`rArr tan(x+y)=(tan x+tany)/(1-tan xtany)=(4tan x)/(1-3tan^(2)x)`
Also `siny=2sinx, cos y=(2)/(3)cosx`
or `sin^(2)y+cos^(2)y=4sin^(2)x+(4cos^(2))/(9)=1`
or `36tan^(2)x+4=9sec^(2)x=9(1+tan^(2)x)`
or `27tan^(2)x=5`
or `tan x=(sqrt(5))/(3sqrt(3))`
`rArr tan(x+y)=((4sqrt(5))/(3sqrt(3)))/(1-(15)/(27))=(4sqrt(5)xx27)/(12xx3sqrt(3))=sqrt(15)`