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Prove that (x^2 + y^2)^4 = (x^4 - 6x^2y^2 + y^4)^2 + (4x^3y -

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Prove that $(x^{2} + y^{2})^{4} = (x^{4} - 6 x^{2} y^{2} + y^{4})^{2} + (4 x^{3} y - 4 x y^{3})^{2}$ .

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Prove that

(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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Text solutionVerified

(x^{2} + y^{2})^{4} = ∣ x + i y ∣^{8}

= ∣ ∣ (x + i y)^{2} ∣ ∣^{4}

= ∣ ∣ (x^{2} - y^{2}) + 2 i x y) ∣ ∣^{4}

= ∣ ∣ [(x^{2} - y^{2}) + 2 i x y)]^{2} ∣ ∣^{2}

= ∣ ∣ (x^{2} - y^{2})^{2} + (2 i x y)^{2} + 2 (x^{2} - y^{2}) (2 i x y) ∣ ∣^{2}

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

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

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

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