First check the stated identity: let x=π/3: sec(x)tan(x)=2√3; -csc(x)cot(x)=-(2/√3)(1/√3)=-⅔. Therefore the identity is false. So we need to solve this as an equation with at least one solution.
sec(x)tan(x)=tan(x)/cos(x)=sin(x)/cos2(x).
-csc(x)cot(x)=-(1/sin(x))cos(x)/sin(x)=-cos(x)/sin2(x).
sin(x)/cos2(x)=-cos(x)/sin2(x),
sin(x)/cos2(x)+cos(x)/sin2(x)=0,
(sin3(x)+cos3(x))/(sin2(x)cos2(x))=0.
Therefore sin3(x)+cos3(x)=0=(sin(x)+cos(x))(sin2(x)-sin(x)cos(x)+cos2(x)),
(sin(x)+cos(x))(1-sin(x)cos(x))=0.
Therefore sin(x)=-cos(x), tan(x)=-1, x=¾π+2πn, 7π/4+2πn, where n is an integer.
sin(x)cos(x)=½sin(2x), so ½sin(2x)=1, sin(2x)=2 is not a solution because sine cannot exceed 1.
So the solution is x=¾π+2πn, 7π/4+2πn.