cot(theta)=cos(theta)/sin(theta), and sec(theta)=1/cos(theta) so sec(theta)cot(theta)=1/sin(theta).
sec^2(theta)cot^2(theta)=1/sin^2(theta).
1/sin^2(theta)-1=(1-sin^2(theta))/sin^2(theta)=cos^2(theta)/sin^2(theta)=cot^2(theta).
Check: let theta=60 degrees; sin60=sqrt(3)/2; cos60=1/2; tan60=sqrt(3); cot60=1/sqrt(3).
sec^2(60)=4; tan^2(60)=3; sin^2(60)=3/4, so sec^2(60)cot^2(60)-1=4/3-1=1/3=cot^2(60).
If sin(theta)=3/5 then tan(theta)=3/4, cot(theta)=4/3 and cos(theta)=4/5.
sec^2(theta)=25/16; cot^2(theta)=16/9; sec^2(theta)cot^2(theta)-1=25/9-1=16/9=cot^2(theta).
Therefore the given identity cannot be true: the left-hand side should be cot^2(theta) not tan^2(theta).