Number of solution of ${\left( {\sin x - 1} \right)^3} + {\left( {\cos x - 1} \right)^3} + {\sin ^3}x = {\left( {2\sin x + \cos x - 2} \right)^3}$
Starting with: $ 2abc + ab\left( {a + b} \right) + bc\left( {b + c} \right) + ac\left( {a + c} \right) = 0$
$$ ab(c + a + b) + bc(a+b+c) + ac\left( {a + c} \right) = 0$$
$$ b(c + a + b)(a+c) + ac\left( {a + c} \right) = 0$$
$$ (a+c)\bigl(b(c + a + b)+ac\bigr) = 0$$ $$ (a+c)(bc + ab + b^2+ac) = 0$$ $$ (a+c)\bigl(c(a+b) + b(a+b)\bigr) = 0$$ $$ (a + b)(a + c)(b + c) = 0$$