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 Th77:
  sin.(th+2 * PI) = sin.th & cos.(th+2 * PI) = cos.th &
  sin.(PI/2-th) = cos.th & cos.(PI/2-th) = sin.th & sin.(PI/2+th) = cos.th &
  cos.(PI/2+th) = -sin.th & sin.(PI+th) = -sin.th & cos.(PI+th) = -cos.th
proof
  thus sin.(th+2 * PI)=sin.(th) * 1 +cos.(th) * 0 by Th73,Th75
    .=sin.th;
  thus cos.(th+2 * PI)=cos.(th) * 1 - sin.(th) * 0 by Th73,Th75
    .=cos.th;
  thus sin.(PI/2-th) =(sin.(PI/2) )* (cos.(-th))
  + (cos.(PI/2)) *( sin.(-th)) by Th73
    .=cos.(th) by Th30,Th75;
  thus cos.(PI/2-th) =(cos.(PI/2)) * (cos.(-th))
  - ( sin.(PI/2)) *( sin.(-th)) by Th73
    .=0 - 1 * (-sin.(th)) by Th30,Th75
    .=sin.(th);
  thus sin.(PI/2+th) =1 * cos.(th) + 0 * (sin.(th)) by Th73,Th75
    .=cos.(th);
  thus cos.(PI/2+th) =(cos.(PI/2)) *( cos.(th))
  -( sin.(PI/2) )*( sin.(th)) by Th73
    .=-sin.(th) by Th75;
  thus
  sin.(PI+th) =(sin.(PI)) * (cos.(th)) +( cos.(PI) )*( sin.(th)) by Th73
    .=-sin.(th) by Th75;
  thus cos.(PI+th) =(-1) * cos.(th) - 0 * sin.(th) by Th73,Th75
    .=-cos.(th);
end;
