Discuss whether the function `f(x)="sin"(cosx+x)` is period or not. If yes, then what is its period?

Explanation

`f(x)=sin(x+cosx)`
Let period be T.
` :. f(x+T)=f(x)` for all real x.
Putting `x=0`, we have
`f(T)=f(0)`
or ` sin(T+cos T)=sin 1`
By intelligent guessing, `T=2pi`, which is the least positive value.
Also, `f(x+2pi)=sin(x+2pi +cos(x+2pi))`
`=sin(2pi+(x+cosx))`
`=sin(x+cosx)=f(x)`