sin(90+θ)=cos(θ); cos(90+θ)=-sin(θ);
tan(90+θ)=sin(90+θ)/cos(90+θ)=cos(θ)/(-sin(θ))=-cot(θ);
cot(90+θ)=-tan(θ);
cosec(90+θ)=1/sin(90+θ)=1/cos(θ)=sec(θ).
The equation can be rewritten:
1/cos(θ)-xcos(θ)sin(θ)/cos(θ)=cos(θ)⇒1/cos(θ)-xsin(θ)=cos(θ),
1-xsin(θ)cos(θ)=cos2(θ),
1-cos2(θ)-xsin(θ)cos(θ)=0,
sin2(θ)=xsin(θ)cos(θ), divide through by sin(θ):
sin(θ)=xcos(θ), x=tan(θ).