Simplify:$(x^2 + y^2 - z^2)^2 - (x^2 - y^2 + z^2)^2$

Given:

$(x^2 + y^2 - z^2)^2 - (x^2 - y^2 + z^2)^2$

To do:

We have to simplify $(x^2 + y^2 - z^2)^2 - (x^2 - y^2 + z^2)^2$.

Solution:

We know that,

$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$

Therefore,

$(x^{2}+y^{2}-z^{2})^{2}-(x^{2}-y^{2}+z^{2})^{2}=[(x^{2})^{2}+(y^{2})^{2}+(-z^{2})^{2}+2 x^{2} y^{2}-2 y^{2} z^{2}-2 z^{2} x^{2}]-[(x^{2})^{2}+(-y^{2})^{2}+(z^{2})^{2}-2 x^{2} y^{2}-2 y^{2} z^{2}+2 z^{2} x^{2}]$

$=x^{4}+y^{4}+z^{4}+2 x^{2} y^{2}-2 y^{2} z^{2}-2 z^{2} x^{2}-x^{4}-y^{4}-z^{4}+2 x^{2} y^{2}+2 y^{2} z^{2}-2 z^{2} x^{2}$

$=4 x^{2} y^{2}-4 z^{2} x^{2}$

$=4 x^{2}(y^{2}-z^{2})$

Hence, $(x^{2}+y^{2}-z^{2})^{2}-(x^{2}-y^{2}+z^{2})^{2}=4 x^{2}(y^{2}-z^{2})$.