Prove that ${cos}^{4} A - {cos}^{2} A = {sin}^{4} A - {sin}^{2} A$

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Solution

${cos}^{4} A - {cos}^{2} A$

$= {cos}^{2} A ({cos}^{2} A - 1)$

$= - {cos}^{2} A (1 - {cos}^{2} A)$

$= - (1 - {sin}^{2} A) {sin}^{2} A$

$= - {sin}^{2} A + {sin}^{4} A$

$= {sin}^{4} A - {sin}^{2} A$

Hence proved.

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