prove (a^2-b^2)^2 + (2ab)^2 = (a^2+b)^2
You asked:
Prove or disprove that the equation $\left(a^{2} - b^{2}\right)^{2} + \left(2 a b\right)^{2} = \left(a^{2} + b\right)^{2}$ is an identity.
MathBot Answer:
The left-hand side of the equation evaluates to:
\[\begin{align*}& \left(a^{2} - b^{2}\right)^{2} + \left(2 a b\right)^{2}\\=& a^{4} + b^{4} - 2 a^{2} b^{2} + \left(2 a b\right)^{2}\\=& a^{4} + b^{4} - 2 a^{2} b^{2} + 4 a^{2} b^{2}\\=& a^{4} + b^{4} + 2 a^{2} b^{2}\\=& a^{4} + 2 a^{2} b^{2} + b^{4}\\\end{align*}\]The right-hand side of the equation evaluates to:
\[\begin{align*}& \left(a^{2} + b\right)^{2}\\=& a^{4} + b^{2} + 2 a^{2} b\\=& a^{4} + 2 a^{2} b + b^{2}\\\end{align*}\]Since $a^{4} + 2 a^{2} b^{2} + b^{4} \ne a^{4} + 2 a^{2} b + b^{2}$, the equation $\left(a^{2} - b^{2}\right)^{2} + \left(2 a b\right)^{2} = \left(a^{2} + b\right)^{2}$ is not an identity.