R1 → R1 - R2 (i.e. Replacing 1^st row by subtraction of 1^st and 2^nd row)

R2 → R2 - R3 (i.e. Replacing 2^nd row by subtraction of 2^nd and 3^rd row)

We know x² - y² = (x + y)(x - y) and y² - z² = (y + z)(y - z)

Therefore taking (x - y) and (y - z) outside the determinant from 1^st and 2^nd row respectively

R1 → R1 - R2 (i.e. Replacing 1^st row by subtraction of 1^st and 2^nd row)

Taking (x - z) outside determinant from 1^st row

C2 → C2 - C3 (i.e. Replacing 2^nd column by subtraction of 2^nd and 3^rd column)

Expanding the determinant along 1^st row

∴ LHS = (x - y)(y - z)(x - z)(z² - xy - xz - yz - z²)

Cancelling z² and adjusting the negative sign with (x - z)

∴LHS = (x - y)(y - z)(z - x)(xy + yz + zx) = RHS

∴