reserve x,x0, r, s, h for Real,

  n for Element of NAT,
  rr, y for set,
  Z for open Subset of REAL,

  f, f1, f2 for PartFunc of REAL,REAL;

theorem Th18:
  cot.(PI/4) = 1 & cot(PI/4) = 1 & cot.(3/4*PI) = -1 & cot(3/4*PI) = -1
proof
A1: sin.(PI/4) <> 0 by Lm9,COMPTRIG:7;
A2: cot.(3/4*PI) = cos.(3/4*PI)*(sin.(3/4*PI))" by Lm10,Th2,RFUNCT_1:def 1
    .= (-sin.(PI/4))/sin.(PI/2 + PI/4) by SIN_COS:78
    .= (-sin.(PI/4))/cos.(PI/4) by SIN_COS:78
    .= -sin.(PI/4)/cos.(PI/4)
    .= -1 by A1,SIN_COS:73,XCMPLX_1:60;
  cot.(PI/4) = cos.(PI/4)/(sin.(PI/4)) by Lm9,Th2,RFUNCT_1:def 1
    .= 1 by A1,SIN_COS:73,XCMPLX_1:60;
  hence thesis by A2,Lm9,Lm10,Th14;
end;
