The solutions of the system of equations `sin x sin y=sqrt(3)/4, cos x cos y= sqrt(3)/4` are

Options

A. `x=pi/3+pi/2 (2n+k), n, k in I`
B. `y=pi/6+pi/2 (k-2n), n, k in I`
C. `x=pi/6+pi/2 (2n+k), n, k in I`
D. `y=pi/3+pi/2 (k-2n), n, k in I`

Answer

A, B, C, D

Explanation

`sin x sin y=sqrt(3)/4` and `cos x cos y = sqrt(3)/4`
Then, `cos x cos y + sin x sin y=sqrt(3)/2`
`rArr cos (x-y)=sqrt(3)/2`
`rArr x-y=2n pi pm pi/6, n in I` ...(i)
and `cos x cos y-sin x sin y=0`
`rArr cos (x+y)=0`
`rArr x+y= k pi +pi/2, k in I` ...(ii)
From Eqs. (i) and (ii), we get
`2x=2npi+k pi pm pi/6+pi/2`
`rArr x=pi/2(2n +k) pm pi/12+pi/4`
`:. x=pi/2 (2n+k)+pi/3 or x=pi/2 (2n+k)+pi/6`
`:. y=pi/6+pi/2 (k-2n) and y=pi/3+pi/2 (k-2n)`