algebra

sin(x) - sin(x) * cos^2(x)  =

sin x  - sin x  *  ( 1 - sin^2 x)  =

sinx  -  sin x  + sin^3 x  =

sin^3 x

sin^4(x) - cos^4(x) divided by sin^2(x) - cos^2(x)

Factor   sin^4 x - cos^4x   as      

[sin^2 x + cos ^2 x] [ sin^2 x - cos^2 x ]  /  [sin^2 x - cos^2 x ]   =

sin^2 x  + cos^2 x   =

1

sin(x) - sin(x) * cos^2(x)

$sin(x) - sin(x) * cos^2(x)\\ =sin(x) (1- cos^2(x))\\ =sin(x) sin^2(x)\\ =sin^3(x)\\ $

sin^4(x) - cos^4(x) divided by sin^2(x) - cos^2(x)

$\frac{sin^4(x) - cos^4(x) }{sin^2(x) - cos^2(x)}\\ =\frac{(sin^2(x) - cos^2(x))(sin^2(x) + cos^2(x)) }{sin^2(x) - cos^2(x)}\\ =\frac{sin^2(x) - cos^2(x)}{sin^2(x) - cos^2(x)}\\~\\ =1 \qquad where \;\;\; x\ne \frac{\pi+2n\pi}{4} \;\;\;n\in Z$