Show that `f(x)=tan^(-1)(sinx+cosx)` is an increasing function on the interval `(0,\ pi//4)` .

Explanation

We have, `f(x) =tan^(-1)(sinx+cosx)`
`therefore f^(')(x) = 1/(1+(sinx+cosx)^(2)).(cosx - sinx)`
`=1/(2+sin2x) (cosx-sinx)` `[therefore sin2x = 2sincosx` and `sin^(2)x+cos^(2)x=1]`
For `f^(')(x) ge0`
`=1/(2+sin2x).(cosx-sinx)ge0`
`rArr cosx - sinx ge0` `[therefore (2+sin2x)ge0 in (0,pi/4)]`
`rArr cosxgesinx`
Which is true, if `x in (0,pi/4)`
Hence, f(x) is an increasing function in `(0,pi/4)`.