Đặt $P$ là biểu thức ban đầu
$\begin{array}{l} {\cot ^2}{30^o}\left( {{{\sin }^8}\alpha - {{\cos }^8}\alpha } \right) = 3\left( {{{\sin }^8}\alpha - {{\cos }^8}\alpha } \right)\\ = 3\left( {{{\sin }^4}\alpha - {{\cos }^4}\alpha } \right)\left( {{{\sin }^4}\alpha + {{\cos }^4}\alpha } \right) = 3\left( {{{\sin }^2}\alpha - {{\cos }^2}\alpha } \right)\left( {{{\sin }^2}\alpha + {{\cos }^2}\alpha } \right)\left( {1 - 2{{\sin }^2}\alpha {{\cos }^2}\alpha } \right)\\ = - 3\cos 2\alpha \left( {1 - \dfrac{1}{2}{{\sin }^2}2\alpha } \right) = - 3\cos 2\alpha + \dfrac{3}{2}{\sin ^2}2\alpha \cos 2\alpha \\ 4\cos {60^o}\left( {{{\cos }^6}\alpha - {{\sin }^6}\alpha } \right) = 2\left( {{{\cos }^2}\alpha - {{\sin }^2}\alpha } \right)\left( {{{\cos }^4}\alpha + {{\sin }^4}\alpha + {{\cos }^2}\alpha {{\sin }^2}\alpha } \right)\\ = 2\cos 2\alpha \left( {1 - {{\sin }^2}\alpha {{\cos }^2}\alpha } \right) = 2\cos 2\alpha - \dfrac{1}{2}\cos 2\alpha {\sin ^2}2\alpha \\ {\sin ^6}\left( {{{90}^o} - \alpha } \right){\left( {{{\tan }^2}\alpha - 1} \right)^3} = {\cos ^6}\alpha .{\left( {\dfrac{1}{{{{\cos }^2}\alpha }} - 2} \right)^3} = {\cos ^6}\alpha \left( {\dfrac{1}{{{{\cos }^6}\alpha }} - \dfrac{6}{{{{\cos }^4}\alpha }} + \dfrac{{12}}{{{{\cos }^2}\alpha }} - 8} \right)\\ = 1 - 6{\cos ^2}\alpha + 12{\cos ^4}\alpha - 8{\cos ^6}\alpha = {\left( {1 - 2{{\cos }^2}\alpha } \right)^3}\\ = - {\cos ^3}2\alpha \\ \Rightarrow P = - \cos 2\alpha + \cos 2\alpha {\sin ^2}2\alpha - \left( { - {{\cos }^3}2\alpha } \right) = \cos 2\alpha \left( {1 - {{\sin }^2}2\alpha } \right) + {\cos ^3}2\alpha \\ = - {\cos ^3}2\alpha + {\cos ^3}2\alpha = 0 \end{array}$
