Firstly, determine the range of the root of the equation: x 2 = | x 2 | = | xsinx+cosx | <| xsinx |+| cosx | < |x|+ 1

Therefore | x | < (1+sqrt(5))/2, which is about 1.6 18.

Let y = x 2-x sinx-cosx.

When x=0, y =- 1

When x increases from 0 to (1+sqrt(5))/2, dy/dx = 2x-sinx-x cosx+sinx = x (2-cosx).

Finally, according to the symmetry, the original equation has only one root in the range of -( 1+sqrt(5))/2 to 0.