$A=\cos^2a+\cos^2a\cot^2a$
$=\cos^2a(1+\cot^2a)$
$=\cos^2a.\Big( 1+\dfrac{\cos^2a}{\sin^2a}\Big)$
$=\cos^2a. \dfrac{\sin^2a+\cos^2a}{\sin^2a}$
$=\dfrac{\cos^2a}{\sin^2a}$
$=\cot^2a$
$B=\sin^2x+\sin^2x\tan^2x$
$=\sin^2x(1+\tan^2x)$
$=\sin^2x\Big(1+\dfrac{\sin^2x}{\cos^2x}\Big)$
$=\sin^2x.\dfrac{\cos^2x+\sin^2x}{\cos^2x}$
$=\dfrac{\sin^2x}{\cos^2x}$
$=\tan^2x$
$C=\dfrac{\sin^2\alpha-\tan^2\alpha}{\cos^2\alpha-\cot^2\alpha}$
$=\dfrac{\sin^2\alpha-\dfrac{\sin^2\alpha}{\cos^2\alpha} }{ \cos^2\alpha-\dfrac{\cos^2\alpha}{\sin^2\alpha} }$
$=\dfrac{ \dfrac{\sin^2\alpha\cos^2\alpha-\sin^2\alpha}{\cos^2\alpha} }{ \dfrac{\cos^2\alpha\sin^2\alpha-\cos^2\alpha}{\sin^2\alpha} }$
$=\dfrac{ \sin^2\alpha.\sin^2\alpha(\cos^2\alpha-1) }{ \cos^2\alpha.\cos^2\alpha(\sin^2\alpha-1)}$
$=\dfrac{\sin^4\alpha.(-\sin^2\alpha) }{\cos^4\alpha.(-\cos^2\alpha)}$
$=\tan^6\alpha$
