STARS wrote: Find the greatest possible value of $ sin(cos x) + cos(sin x)$ and determine all real numbers x, for which this value is achieved. $ Max(Sin(Cos(x))) = Sin(1)$ -> is for $ x = 2{\pi}N$ ( N - integer ) $ Max(Cos(Sin(x))) = 1$ -> is for $ x = {\pi}K$ ( K - integer ) So : $ Max(Cos(Sin(x))+ Sin(Cos(x)))= 1 + Sin(1)$ -> is for $ x = 2{\pi}N$ ( N - integer )

let $ f : x==>sin(cos(x))+cos(sin(x))$. $ f$ is periodic with period $ 2\pi$ .and $ f(-x)=f(x)$. in $ [0,\pi]$ we have $ f$ is non increasing (if $ x<y$ then $ f(x)>f(y)$) in $ [0,\pi/2]$ so $ f(x) \le f(0)= sin(1)+1$ and for $ [\pi/2,\pi] f(x) <1$.so the maximal value is $ 1+sin(1)$ which is achieved when $ x=0$.

see here http://www.mathlinks.ro/viewtopic.php?t=215123