if x and y are acute angles such that `cosx + cosy = 3/2` and `sinx + siny = 3/4` then `sin(x+y)=`

Options

A. `(2)/(5)`
B. `(3)/(4)`
C. `(3)/(5)`
D. `(4)/(5)`

Answer

D

Explanation

Given, `cosx+cosy=(3)/(2)`
`implies 2cos((x+y)/(2))cos.(x-y)/(2)=(3)/(2)" ... (i)"`
and `sinx+siny=(3)/(4)`
`implies2sin((x+y)/(2))cos.((x-y)/(2))=(3)/(4)" ... (ii)"`
On dividing Eq. (i) by Eq. (ii), we get
`(2sin((x+y)/(2))cos.((x-y)/(2)))/(2cos.((x+y)/(2))cos.((x-y)/(2)))=(3//4)/(3//2)`
`implies tan.((x+y)/(2))=(1)/(2)`
`therefore sin(x+y)=(2tan.((x+y)/(2)))/(1+tan^(2)((x+y)/(2)))`
`=(2xx(1)/(2))/(1+((1)/(2))^(2))=(4)/(4+1)=(4)/(5)`