reserve y for set,
  x,r,a,b for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2,f3 for PartFunc of REAL,REAL;

theorem Th46:
  cos.x<>0 implies sin/cos is_differentiable_in x & diff(sin/cos,
  x)=1/(cos.x)^2
proof
  assume
A1: cos.x<>0;
A2: sin is_differentiable_in x & cos is_differentiable_in x by SIN_COS:63,64;
  then
  diff(sin/cos,x) =(diff(sin,x) * cos.x - diff(cos,x)*sin.x)/(cos.x)^2 by A1,
FDIFF_2:14
    .=((cos.x)*cos.x - diff(cos,x)*sin.x)/(cos.x)^2 by SIN_COS:64
    .=((cos.x)*(cos.x)-(-sin.x)*(sin.x))/(cos.x)^2 by SIN_COS:63
    .=((cos.x)*(cos.x) + (sin.x)*(sin.x))/(cos.x)^2
    .=1/(cos.x)^2 by SIN_COS:28;
  hence thesis by A2,A1,FDIFF_2:14;
end;
