$\frac{1}{{{{\cos }^4}\alpha }} - \frac{1}{{{{\cot }^4}\alpha }} = \frac{1}{{{{\cos }^2}\alpha }} + \frac{1}{{{{\cot }^2}\alpha }} = \frac{1}{{{{\cos }^2}\alpha }} + \frac{{{{\sin }^2}\alpha }}{{{{\cos }^2}\alpha }}$

$ = \frac{{{{\cot }^4}\alpha  - {{\cos }^4}\alpha }}{{{{\cos }^4}\alpha {{\cot }^4}\alpha }} = \frac{{({{\cot }^2}\alpha  - {{\cos }^2}\alpha )({{\cot }^2}\alpha  + {{\cos }^2}\alpha )}}{{{{(\cos \alpha \cot \alpha )}^4}}} = \frac{{(\frac{{{{\cos }^4}\alpha }}{{{{\sin }^2}\alpha }} - {{\cos }^2}\alpha )(\frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} + {{\cos }^2}\alpha )}}{{{{(\frac{{{{\cos }^2}\alpha }}{{\sin \alpha }})}^2}}}$

$ = \frac{{{{\cos }^2}\alpha (\frac{1}{{{{\sin }^2}\alpha }} - 1){{\cos }^2}\alpha (\frac{1}{{{{\sin }^2}\alpha }} + 1)}}{{{{\cos }^4}\alpha  \times \frac{1}{{{{\sin }^2}\alpha }}}} = \frac{{{{\cos }^4}\alpha (\frac{1}{{{{\sin }^2}\alpha }} + 1)(\frac{1}{{{{\sin }^2}\alpha }} + 1)}}{{{{\cos }^4}\alpha  \times \frac{1}{{{{\sin }^2}\alpha }}}}$

$ = \frac{{\frac{1}{{{{\sin }^4}\alpha }} - 1}}{{\frac{1}{{{{\sin }^2}\alpha }}}} = \frac{{\frac{{1 - {{\sin }^4}\alpha }}{{{{\sin }^4}\alpha }}}}{{\frac{1}{{{{\sin }^2}\alpha }}}} = \frac{{(1 - {{\sin }^4}\alpha ){{\sin }^2}\alpha }}{{{{\sin }^4}\alpha }} = \frac{{1 - {{\sin }^4}\alpha }}{{{{\sin }^2}\alpha }}$

$ \Rightarrow \frac{{(1 - {{\sin }^2}\alpha )(1 + {{\sin }^2}\alpha )}}{{{{\sin }^2}\alpha }} = \frac{{{{\cos }^2}\alpha (1 + {{\sin }^2}\alpha )}}{{{{\sin }^2}\alpha }} = {\cot ^2}\alpha (1 + {\sin ^2}\alpha )$

$ = {\cot ^2}\alpha  + {\cos ^2}\alpha  = \frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} + {\cos ^2}\alpha  = {\cos ^2}\alpha (\frac{1}{{{{\sin }^2}\alpha }} + 1) = {\cos ^2}\alpha (\frac{{1 + {{\sin }^2}\alpha }}{{{{\sin }^2}\alpha }})$