The total number of solution of `sin^(4)x+cos^(4) x= sin x cos x` in `[0, 2pi]`

Options

A. 2
B. 4
C. 6
D. none of these

Answer

A

Explanation

`sin^(4) x+cos^(4)x=sin x cos x`
or `(sin^(2) x+cos^(2) x)^(2)-2 sin^(2) x cos^(2) x= sin x cos x`
or `1- (sin^(2) 2x)/2=(sin 2x)/2`
or `sin^(2) 2x+sin 2x-2 =0`
or `(sin 2x+2) (sin 2x-1)=0`
or `sin 2x=1`
or `2x=(4n+1) pi/2, n in Z`
or `x=(4n+1) pi/4, n in Z`
`=pi/4, (5pi)/4" "( :' x in [0, 2pi])`
Thus, there are two solutions.