reserve x,x1,x2,x3 for Real;

theorem Th7:
  cos(2*x)=(cos(x))^2-(sin(x))^2 & cos(2*x)=2*(cos(x))^2-1 & cos(2*
  x)=1-2*(sin(x))^2
proof
A1: cos(2*x) = cos(x+x) .=(cos(x))^2 -(sin(x))^2 by SIN_COS:75;
  then cos(2*x)=(cos(x))^2 -(sin(x))^2-1+1
    .=(cos(x))^2 -(sin(x))^2-((cos(x))^2+(sin(x))^2)+1 by SIN_COS:29
    .=-(-1+ 2*(sin(x))^2);
  hence thesis by A1;
end;
