Given: $cos x + {cos}^{2} x = 1$
Prove that:
${sin}^{2} x + {sin}^{4} x = 1$

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Solution

$cosx + {cos}^{2} x = 1$
$cosx = 1 - {cos}^{2} x$
$cos x = {sin}^{2} x$
${sin}^{4} x - {cos}^{2} x = 0$
${sin}^{4} x + 1 - {cos}^{2} x = 1$
$∴ {sin}^{4} x + {sin}^{2} x = 1$

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