Remember the following rule for factoring a difference of cubes:
[tex]a^3-b^3=(a-b)(a^2+ab+b^2)[/tex]Notice that 1 is equal to 1^3.
Then, to factor the expression:
[tex]1-\sin ^3x[/tex]Use the identity above with a=1 and b=sin(x):
[tex]\begin{gathered} 1-\sin ^3x=(1-\sin x)(1^2+1\cdot\sin x+\sin ^2x) \\ =(1-\sin x)(1+\sin x+\sin ^2x) \end{gathered}[/tex]Therefore, a factored form of the given expression is:
[tex]1-\sin ^3x=(1-\sin x)(1+\sin x+\sin ^2x)[/tex]