The number of integers in the range of the function
`f(x)=|4((sqrt(cosx)-sqrt(sinx))(sqrt(cosx)+sqrt(sinx)))/((cosx+sinx))|` is ________.

Answer

5

Explanation

`f(x)=|4((sqrt(cosx)-sqrt(sinx))(sqrt(cosx)+sqrt(sinx)))/((cosx+sinx))|` is defined only if
`cosx gt 0, sinx gt 0`
Therefore, x lies in first quadrant only.
`f(x)=|4((cosx-sinx))/((cosx+sinx))|=|4tan((pi)/(4)-x)|`
`=|4tan(x-(pi)/(4))|`
Now, `0 le x le (pi)/(2)`
`or (-pi)/(4) le x-(pi)/(4) le (pi)/(4)`
`or -1 le tan(x-(pi)/(4)) le 1`
` or -4 le 4 tan(x-(pi)/(4))le 4`
`0 le |4tan(x-(pi)/(4))| le 4`