For `0ltalphaltbetalt(pi)/(2),if(sinalpha)/(sinbeta)=(sqrt3)/(2)and (cosalpha)/(cosbeta)=(sqrt5)/(2),then`

Options

A. `tan alpha =1`
B. `tan beta=(sqrt3)/(sqrt5)`
C. `tan beta=1`
D. `tan alpha=(sqrt3)/(sqrt5)`

Explanation

We have, `(sinalpha)/(sinbeta).(cosbeta)/(cosalpha)=(sqrt3)/(2).(2)/(sqrt5)=(sqrt3)/(sqrt5)=(tanalpha)/(tanbeta)implies(tanalpha)/(sqrt3)=(tanbeta)/(sqrt5)=k(say)`
`Also, 2 sinalpha=sqrt3sinbetaimplies(2tanalpha)/(sqrt(1+tan^(2)alpha))=(sqrt2tanbeta)/(sqrt(a+tan^(2)beta))`
`implies(2 sqrt3k)/(sqrt(1+3k^(2)))=(sqrt3.sqrt5k)/(sqrt(1+5k^(2)))implies4(1+5k^(2))=5(1+3k^(2))`
`implies5k^(2)=1impliesk=(1)/(sqrt5)`
Hence, `tanalpha=(sqrt3)/(sqrt5)and tanbeta=1.`