a1 = 
      (1/Pi)*(∫_(-Pi)^Pi) Cos[x - (Pi/4)]*
            Cos[x] \[DifferentialD]x

b1 = 
      (1/Pi)*(∫_(-Pi)^Pi) Cos[x - (Pi/4)]*
            Sin[x] \[DifferentialD]x

(Subscript[c, 1]) = 
      (1/(2*Pi))*(∫_(-Pi)^Pi) Cos[
              x - (Pi/4)]*(E^(-I*1*
                  x)) \[DifferentialD]x

(((1/2) - (I/2))/(Sqrt[2]))
(Subscript[c, -1]) = 
      (1/(2*Pi))*(∫_(-Pi)^Pi) Cos[
              x - (Pi/4)]*(E^(I*1*
                  x)) \[DifferentialD]x

(((1/2) + (I/2))/(Sqrt[2]))

Pi*(a1^2) + Pi*(b1^2)

2*Pi*(((1/2) - (I/2))/(Sqrt[2]))*(((1/2) + (I/2))/(Sqrt[2])) + 2*Pi*(((1/2) + (I/2))/(Sqrt[2]))*(((1/2) - (I/2))/(Sqrt[2]))

(∫_0^(2*Pi))Cos[
              x - (Pi/4)]^2 \[DifferentialD]x