Number of solution of ${\left( {\sin x - 1} \right)^3} + {\left( {\cos x - 1} \right)^3} + {\sin ^3}x = {\left( {2\sin x + \cos x - 2} \right)^3}$

Question

Number of solution of the equation ${\left( {\sin x - 1} \right)^3} + {\left( {\cos x - 1} \right)^3} + {\sin ^3}x = {\left( {2\sin x + \cos x - 2} \right)^3}$ in the interval $[0,2\pi]$ is equal to_____

My approach is as follow

$a = \sin x - 1;b = \cos x - 1;c = \sin x$

${a^3} + {b^3} + {c^3} = {\left( {a + b + c} \right)^3}$

$ \Rightarrow {a^3} + {b^3} + {c^3} = {\left( {a + b + c} \right)^3}$

${\left( {a + b + c} \right)^3} = {a^3} + {b^3} + {c^3} + 6abc + 3ab\left( {a + b} \right) + 3bc\left( {b + c} \right) + 3ac\left( {a + c} \right)$

${a^3} + {b^3} + {c^3} = {\left( {a + b + c} \right)^3} \Rightarrow 6abc + 3ab\left( {a + b} \right) + 3bc\left( {b + c} \right) + 3ac\left( {a + c} \right) = 0$

$ \Rightarrow 2abc + ab\left( {a + b} \right) + bc\left( {b + c} \right) + ac\left( {a + c} \right) = 0$

How do we approach from here

Answer 1

Starting with: $ 2abc + ab\left( {a + b} \right) + bc\left( {b + c} \right) + ac\left( {a + c} \right) = 0$

$$ ab(c + a + b) + bc(a+b+c) + ac\left( {a + c} \right) = 0$$

$$ b(c + a + b)(a+c) + ac\left( {a + c} \right) = 0$$

$$ (a+c)\bigl(b(c + a + b)+ac\bigr) = 0$$ $$ (a+c)(bc + ab + b^2+ac) = 0$$ $$ (a+c)\bigl(c(a+b) + b(a+b)\bigr) = 0$$ $$ (a + b)(a + c)(b + c) = 0$$

Answer 2

Having $a = \sin x - 1$ and $c = \sin x$ is redundant, it hides potential cancellation and simplification, and leaves you trying to factor a polynomial in three variables instead of just two. Instead, everywhere you have a $c$, replace it with $a+1$. \begin{align*} a^3 + b^3 + (a+1)^3 &= (a+b+a+1)^3 \\ a^3 + b^3 + (a+1)^3 &= (2a+b+1)^3 \\ 1 + 3a + 3a^2 + 2a^3 + b^3 &= 8 a^3+12 a^2 b+12 a^2+6 a b^2+12 a b+6 a+b^3+3 b^2+3 b+1 \end{align*}\begin{align*} -6 a^3-12 a^2 b-9 a^2-6 a b^2-12 a b-3 a-3 b^2-3 b &= 0 \\ -3(2 a^3+4 a^2 b+3 a^2+2 a b^2+4 a b+a+b^2+b) &= 0 \\ -3(2 a^3 + 2 a^2 b + 2 a^2 b + 2 a b^2 + 3 a^2 + 3 a b + a b + b^2 + a+b) &= 0 \\ -3(a+b)(2 a^2 + 2 a b + 3 a + b + 1) &= 0 \\ -3(a+b)(2 a^2 + 2 a b + 2 a + a + b + 1) &= 0 \\ -3(a+b)(a+b+1)(2 a + 1) &= 0 \end{align*}

Note that the factor $a+b+1 = \sin x + \cos x - 1$ leads to three solutions in $[0,2\pi]$.

Answer 3

Hint

$$(a+b+c)^3-c^3=(a+b+c-c)((a+b+c)^2+(a+b+c)c+c^2)$$

$$a^3+b^3=?$$

Clearly $a+b$ is a common factor.

By symmetry, we can claim that the other two factors of $$(a+b+c)^3-a^3-b^3-c^3$$ will be $b+c,c+a$

Alternatively simplify

$$((a+b+c)^2+(a+b+c)c+c^2)-(a^2-ab+b^2)$$