Since x and y are form 0 to Pi/2, eliminating Cos and convering Sin[x] == sx, Sin[y] == sy.

Since x and y are form 0 to Pi/2, eliminating Cos and converting Sin[x] == sx, Sin[y] == sy.

A solution with Reduce

Didn't get Maximize to extract max of an from red.

Edit

Solution with Maximize.

Maximize works best with linear and polynomial expressions.( See help function) Therefore using variable transformations.

an = x^n - x^(2 n) // Factor

(*   -x^n (-1 + x^n)   *)

max = Maximize[{an /. x^n -> xn, 0 <= xn <= 1}, xn]

(*   {1/4, {xn -> 1/2}}   *)

Solve[(xn /. max[[2]]) == x^n && 0 <= x <= 1 && n > 0, x]

(*   {{x -> ConditionalExpression[2^(-1/n), n > 0]}}   *)

A solution with Reduce

Didn't get Maximize to extract max of an from red.

Example from comment.

Since x and y are form 0 to Pi/2, eliminating Cos and convering Sin[x] == sx, Sin[y] == sy.

anequ = an==n(Sin[x] + Cos[y] + Cos[x - y])//TrigExpand

(*   (0 <= x <= \[Pi]/2, 0 <= y <= \[Pi]/2)    *)

eli = Eliminate[{anequ, Sin[x]^2 + Cos[x]^2 == 1, 
   Sin[y]^2 + Cos[y]^2 == 1, Sin[x] == sx, Sin[y] == sy}, {Cos[x], 
   Cos[y]}]

sol1 = First@
   Solve[{Sin[x] == sx, Sin[y] == sy, 0 <= x <= Pi/2, 
     0 <= y <= Pi/2}, {x, y}] // 
  Simplify[#, {0 <= x <= Pi/2, 0 <= y <= Pi/2}] &

eli2 = List @@ (eli /. sol1 // Simplify)

(*   {an^4 + 4 n^4 sx^2 (1 + sy)^2 (-1 + sx^2 + sy^2) + 
   4 an^2 n^2 (1 + sy) (-1 + sy + sx^2 (2 + sy)) == 
  4 an n sx (1 + sy) (an^2 + 2 n^2 (1 + sy) (-1 + sx^2 + sy)), 
 0 <= sy < 1 && 0 <= sx < 1}   *)

max = Maximize[{an, eli2 && n > 0}, {an, sx, sy}] // 
  Simplify[#, Assumptions -> Element[n, Integers] && n > 0] &

(*   {(3 Sqrt[3] n)/2, {an -> (3 Sqrt[3] n)/2, sx -> Sqrt[3]/2, sy -> 1/2}}   *)

{x -> ArcSin[sx], y -> ArcSin[sy]} /. max[[2]]

(*   {x -> Pi/3, y -> Pi/6}   *)