For `0ltxlt(pi)/(2)`,prove that `x gt sin x `and hence ` cos (sin x) gt sin (cosx) `

Explanation

Let f(x) =x-sinx
`f(x) =1 -cosx gt 0,forall in (0,pi//2)`
So ,f(x) is an increasing function.
for `x in (0,pi//2),f(x)gt0`
or x-`sinxgt0`
`xgt sin x`
Now `cos x in (0,1) for x in (0,pi//2)`
So, from (1), we have (replacing x by cosx)
cosx `gt` sin (cosx)
Also ,cos x is decreasing in `(0,pi//2)`
Therefore from (1),we have (applying cos both sides)
`cos lt cos (sin x)`
from (2) and (3) we get
`sin (cos x) lt cos xltcos(sin x)`
or `cos (sin x)gt sin (cosx)`