What are the values of (x, y) satisfying the simultaneous equation `sin^(-1)x + sin^(-1)y=(2pi)/(3)and cos^(-1)x - cos^(-1)y=(pi)/(3)` ?

Options

A. (0, 1)
B. `((1)/(2),1)`
C. `(1, (1)/(2))`
D. `((sqrt(3))/(2), 1)`

Answer

B

Explanation

Given that
`sin^(-1)x + sin^(-1)y=(2pi)/(3)" "...(i)`
and `cos^(-1)x - cos^(-1)y=(pi)/(3)`
`rArr (pi)/(2)-sin^(-1)x-(pi)/(2)+sin^(-1)y=(pi)/(3)" "["since" cos^(-1)x + sin^(-1)x=(pi)/(2)]`
`rArr - sin^(-1)x + sin^(-1)y = (pi)/(3)" "...(ii)`
On solving eq. (i) and (ii), we get
`2 sin^(-1)y=pi and 2 sin^(-1)x = (pi)/(3)`
`rArr sin^(-1)y = (pi)/(2) and sin^(-1)x=(pi)/(6)`
Hence, `y = "sin"(pi)/(2) and x = "sin"(pi)/(6)`
`rArr x = (1)/(2) and y = 1`