$Assumptions={n,k}∈Integers;

The Fourier Expansion of a Triangle Wave in a Non-Standard Domain

triangle[x_]:=-4Abs[x]+1;
fparams={1,2Pi}; (* Period 1 *)

Manipulate[
Plot[
{triangle[x],Evaluate[FourierSeries[triangle[x],x,iter,FourierParameters->fparams]]},
{x,-1/2,1/2}],
{{iter,1,"FS Iteration Limit"},1,10,1,Appearance->"Labeled"}]

Trigonometric Simplifications for Integer Multiples of Pi

DiscretePlot[{Sin[n Pi],Cos[n Pi]},{n,1,10},ExtentSize->Full,PlotLegends->"Expressions"]

Real Fourier Coefficients (Discrete Sine and Cosine, with Modified Parameters)

Simplify[FourierCosCoefficient[triangle[x],x,n,
FourierParameters->fparams],Assumptions->n≠0]
(* a_0 = 2 - 4 Pi *)

Simplify[FourierSinCoefficient[triangle[x],x,2k+1,
FourierParameters->fparams]]

Complex Fourier Coefficients (Integrated Exponential Form, with Canonical Parameters)

Simplify[FourierCoefficient[triangle[x],x,n],Assumptions->n≠0]
(* c_0 = 1 - 2 Pi *)