Min value of cos(sinx)+sin(cosx) The max value of $f(x) = \sin(\cos x) + \cos(\sin x)$ is equal to $1 + \sin 1$.  What is the minimum value of $f(x)$?

I found the max value using that range $\sin(\cos x)$ is $[- \sin 1, \sin 1]$ and range of $\cos(\sin x)$ is $[ \cos 1, 1]$ and the maximum of both occurs at $x=0$.  Although I can't find the min value using this approach. Please try to find this  without using graphing calculators.
Look at this graph of your function to see the multiple minima, essential for understanding and solving your problem.
Look at this graph of your function to see the multiple minima, essential for understanding and solving your problem.