reserve q,th,r for Real,
  a,b,p for Real,
  w,z for Complex,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,cq1 for Complex_Sequence,
  rseq,rseq1,rseq2 for Real_Sequence,
  rr for set,
  hy1 for 0-convergent non-zero Real_Sequence;
reserve d for Real;
reserve th,th1,th2 for Real;

theorem Th75:
  cos.(PI/2) = 0 & sin.(PI/2) = 1 & cos.(PI) = -1 & sin.(PI) = 0 &
  cos.(PI+PI/2) = 0 & sin.(PI+PI/2) = -1 & cos.(2 * PI) = 1 & sin.(2 * PI) =0
proof
  thus
A1: cos.(PI/2)=cos.(PI/4+PI/4) .=(cos.(PI/4)) * (cos.(PI/4))
  -(cos.(PI/4)) * (cos.(PI/4)) by Th72,Th73
    .=0;
  thus
A2: sin.(PI/2)=sin.(PI/4+PI/4) .= (cos.(PI/4)) *( cos.(PI/4))
  +(sin.(PI/4)) * (sin.(PI/4)) by Th72,Th73
    .=1 by Th28;
  thus
A3: cos.(PI)=cos.(PI/2+PI/2) .=0 * 0-sin.(PI/2) * sin.(PI/2) by A1,Th73
    .=-1 by A2;
  thus
A4: sin.(PI)=sin.(PI/2+PI/2) .=(sin.(PI/2)) *( cos.(PI/2))
  +(cos.(PI/2) )*( sin.(PI/2)) by Th73
    .=0 by A1;
  thus cos.(PI+PI/2) =(cos.(PI)) *( cos.(PI/2))
  -(sin.(PI)) * (sin.(PI/2)) by Th73
    .=0 by A1,A4;
  thus sin.(PI+PI/2) =(sin.(PI) )*( cos.(PI/2))
  +(cos.(PI)) *( sin.(PI/2) ) by Th73
    .=-1 by A1,A2,A3;
  thus cos.(2 * PI)=cos.(PI+PI) .=(-1 ) * (-1) -sin.(PI) * sin.(PI) by A3,Th73
    .=1 by A4;
  thus sin.(2 * PI)=sin.(PI+PI)
    .=(sin.(PI)) * (cos.(PI))+(cos.(PI)) * (sin.(PI)) by Th73
    .=0 by A4;
end;
