If `sin^4x + cos^4y + 2 = 4 sin x.cos y` and `0 le x, y le pi/2` then sin x + cos y is equal to

Options

A. `-2`
B. 0
C. 2
D. none of these

Answer

C

Explanation

We have,
`"sin"^(4)x + "cos"^(4)y + 2 = 4 "sin" x"cos" y`
`rArr ("sin"^(2)x-1)^(2) + ("cos"^(2)y-1)^(2) + 2("sin"^(2)x + "sin"^(2)y)-4"sin"x "cos"y=0`
`rArr ("sin"^(2)x-1)^(2) + ("cos"^(2)y-1)^(2) + 2("sin"x -"cos"y)^(2) = 0`
`rArr "sin"^(2) x-1 = 0, "cos"^(2)y-1 = 0 "and sin"x -"cos" y = 0 " " [because 0 le x, y le pi//2]`
`rArr "sin" x = 1 "and cos" y = 1`
`rArr "sin" x + "cos"y = 2`