c)
$\begin{array}{l} C = \dfrac{{{{\cot }^2}x - {{\cos }^2}x}}{{{{\cot }^2}x}} + \dfrac{{\sin x\cos x}}{{\cot x}}\\ C = 1 - {\cos ^2}x.\dfrac{1}{{{{\cot }^2}x}} + \sin x\cos x.\tan x\\ C = 1 - {\cos ^2}x.{\tan ^2}x + \sin x.\cos x.\dfrac{{\sin x}}{{\cos x}}\\ C = 1 - {\sin ^2}x + {\sin ^2}x = 1 \end{array}$
d)
$\begin{array}{l} D = {\sin ^6}x + {\cos ^6}x - 2{\sin ^4}x - {\cos ^4}x + {\sin ^2}x\\ D = \left( {{{\sin }^2}x + {{\cos }^2}x} \right)\left( {{{\sin }^4}x - {{\sin }^2}x{{\cos }^2}x + {{\cos }^4}x} \right) - 2{\sin ^4}x - {\cos ^4}x + {\sin ^2}x\\ D = {\sin ^4}x + {\cos ^4}x - {\sin ^2}x{\cos ^2}x - 2{\sin ^4}x - {\cos ^4}x + {\sin ^2}x\\ D = - {\sin ^4}x - {\sin ^2}x{\cos ^2}x + {\sin ^2}x\\ D = \left( { - {{\sin }^2}x{{\cos }^2}x + {{\sin }^2}x} \right) - {\sin ^4}x\\ D = {\sin ^2}x\left( {1 - {{\cos }^2}x} \right) - {\sin ^4}x\\ D = {\sin ^4}x - {\sin ^4}x = 0 \end{array}$
