Partendo dal lato sinistro (LHS) dell'identità

Applicare la formula: $a^3+b$$=\left(a+\sqrt[3]{b}\right)\left(a^2-a\sqrt[3]{b}+\sqrt[3]{b^{2}}\right)$, dove $a=\sin\left(x\right)$ e $b=1$

Applicare la formula: $ab$$=ab$, dove $ab=- 1\sin\left(x\right)$, $a=-1$ e $b=1$

Applicare la formula: $\frac{a}{a}$$=1$, dove $a=\sin\left(x\right)+1$ e $a/a=\frac{\left(\sin\left(x\right)+1\right)\left(\sin\left(x\right)^2-\sin\left(x\right)+1\right)}{\sin\left(x\right)+1}$

Since we have reached the expression of our goal, we have proven the identity