t a n 2 α − t a n α c o t α − c o t 2 α = t a n 3 α c o t α = 1 t a n α / ( ) 2 c o t 2 α = 1 t a n 2 α ↓ c o t α − c o t 2 α = 1 t a n α − 1 t a n 2 α c o t α − c o t 2 α = t a n α − 1 t a n 2 α t a n 2 α − t a n α t a n α − 1 t a n 2 α = t a n 3 α t a n 2 α ( t a n 2 α − t a n α ) t a n α − 1 = t a n 3 α t a n 2 α ( t a n 2 α − t a n α ) = t a n 3 α ( t a n α − 1 ) t a n 2 α − t a n α = t a n α ( t a n α − 1 ) t a n 2 α − t a n α = t a n 2 α − t a n α {\displaystyle {\begin{aligned}{\frac {tan^{2}\alpha -tan\alpha }{cot\alpha -cot^{2}\alpha }}=tan^{3}\alpha \\&&cot\alpha ={\frac {1}{tan\alpha }}/()^{2}\\&&cot^{2}\alpha ={\frac {1}{tan^{2}\alpha }}\\&&\downarrow \\&&cot\alpha -cot^{2}\alpha ={\frac {1}{tan\alpha }}-{\frac {1}{tan^{2}\alpha }}\\&&cot\alpha -cot^{2}\alpha ={\frac {tan\alpha -1}{tan^{2}\alpha }}\\{\frac {tan^{2}\alpha -tan\alpha }{\frac {tan\alpha -1}{tan^{2}\alpha }}}=tan^{3}\alpha \\{\frac {tan^{2}\alpha (tan^{2}\alpha -tan\alpha )}{tan\alpha -1}}=tan^{3}\alpha \\tan^{2}\alpha (tan^{2}\alpha -tan\alpha )=tan^{3}\alpha (tan\alpha -1)\\tan^{2}\alpha -tan\alpha =tan\alpha (tan\alpha -1)\\tan^{2}\alpha -tan\alpha =tan^{2}\alpha -tan\alpha \end{aligned}}}