Prove the following identities :
$(x^{2} + a^{2})^{4} + (x^{4} - 6 x^{2} a^{2} + a^{4})^{2} + (4 x^{3} a - 4 x a^{3})^{2}$

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Solution

We have $(x^{2} + a^{2})^{4} = (x + a i)^{4} (x - a i)^{4}$
Now $(x + a i)^{4} = x^{4} + 4 x^{3} a i + 6 x^{2} a^{2} i^{2} + 4 x a^{3} i^{3} + a^{4} i^{4}$
$= (x^{4} - 6 x^{2} a^{2} + a^{4}) + i (4 x^{3} a - 4 x a^{3})$ ..... (1)
Similarly,
$(x - a i)^{4} = (x^{4} - 6 x^{2} a^{2} + a^{4}) - i (4 x^{3} a - 4 x a^{3})$ ..... (2)
Multiplying (1) and (2) we get,
$(x + a i)^{4} (x - a i)^{4} = (x^{4} - 6 x^{2} a^{2} + a^{4})^{2} + (4 x^{3} a - 4 x a^{3})^{2}$

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(x^{2} + y^{2})^{4} = (x^{4} - 6 x^{2} y^{2} + y^{4})^{2} + (4 x^{3} y - 4 x y^{3})^{2}

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