If `0 le x le (pi)/(2)` and `sin^(2)x=a` and `cos^(2)x=b`, then `sin 2x+cos 2x=`

Options

A. `2sqrt(ab)+b-a`
B. `2ab+a^(2)-b^(2)`
C. `2sqrt(ab)+2a-1`
D. `2ab+2a-1`

Answer

A

Explanation

Start by finding the positive square roots of `sin^(2)x` and `cos^(2)x` :
`sin x = sqrt(a)`
`cos x = sqrt(b)`
Next, use the double angle formula to simplify sin 2x and cos 2x :
`sin 2x = 2 sin x cos x = 2sqrt(ab)`
`cos 2x = cos^(2)x - sin^(2)x = b-a`
Therefore, `sin 2x + cos 2x = 2sqrt(ab)+b-a`.