If `sin^(4)alpha+cos^(4)beta+2=4 sin alpha cos beta, 0 le alpha, (pi)/(2)`, then `(sin alpha+cos beta)` is equal to

Options

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

Answer

C

Explanation

`sin^(4)alpha+cos^(4)beta+2=4 sin alpha cos beta`
`rArr (sin^(2)alpha-1)^(2)+(cos^(2)beta-1)^(2)+2(sin alpha-cos beta)^(2)=0`
`rArr sin^(2)alpha=1, cos^(2)beta = 1` and `sin alpha = cos beta`
Now `0 le alpha, beta le (pi)/(2)`
`therefore sin alpha = 1` and `cos beta = 1`
Hence, `(sin alpha +cos beta) = 1+1=2`