[tex]\stackrel{\textit{Pythagorean Identities}}{1+tan^2(\theta)=sec^2(\theta)}\implies tan^2(\theta ) = sec^2(\theta )-1 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ csc^2(\theta )tan^2(\theta )-1~~ = ~~tan^2(\theta ) \\\\[-0.35em] ~\dotfill[/tex]
[tex]csc^2(\theta )tan^2(\theta )-1\implies \cfrac{1}{sin^2(\theta )}\cdot \cfrac{sin^2(\theta )}{cos^2(\theta )}-1\implies \cfrac{1}{cos^2(\theta )}-1 \\\\\\ \cfrac{1^2}{cos^2(\theta )}-1\implies \left[ \cfrac{1}{cos(\theta )}\right]^2-1\implies sec^2(\theta )-1\implies tan^2(\theta )[/tex]
