The set of values of 'a' for which the equation `"sin" x ("sin"x +"cos" x) = a` has real solutions, is

Options

A. `[1-sqrt(2), 1+sqrt(2)]`
B. `[2-sqrt(3), 2+sqrt(3)]`
C. `[0, 2+sqrt(3)]`
D. `[(1-sqrt(2))/(2), (1+sqrt(2))/(2)]`

Answer

D

Explanation

We have,
`"sin" x ("sin" x + "cos" x) = a`
`rArr 1 -"cos" 2x + "sin" 2x = 2a`
`rArr "sin" 2x - "cos" 2x = 2a - 1`
This equation will have real solutions, if
`|2a-1| lt sqrt(1+1)`
`rArr 1-sqrt(2) lt 2a lt 1 + sqrt(2) rArr a in [ (1-sqrt(2))/(2), (1+sqrt(2))/(2)]`