Also:

SetSystemOptions[
  "SimplificationOptions" -> "AutosimplifyTrigs" -> False];

Then either of these:

Simplify[D[Sin[x] - (x - Sqrt[2]/2 x Sin[x])/Cos[x], x], 
 ComplexityFunction -> (LeafCount[#] + 
     10 Count[#, _Sec | _Csc | _Cot, Infinity] &)]

D[Sin[x] - (x - Sqrt[2]/2 x Sin[x])/Cos[x], x] // Simplify // 
 ReplaceAll[Sec[t_] :> 1/Cos[t]]

Simplify[] will convert 1/Cos[t] to Sec[t], since Sec[t] is less complex. You have to modify the complexity function to indicate to Simplify[] that you consider Sec[t] more complex than 1/Cos[t].

Or manually substitute the desired forms for the undesired ones. With "AutosimplifyTrigs" turned off, they stay as substituted.

Although the obtained result does not contain secant terms, its form is not what I wanted. The desired form is as shown in the image in the original post.

Thank you for your help. This result is exactly in the format I wanted.