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

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Solution

$\begin{matrix} L H S = {sin}^{4} x + {cos}^{4} x = {({sin}^{2} x)}^{2} + {cos}^{4} x = {(1 - {cos}^{2} x)}^{2} + {cos}^{4} x = 1 + {cos}^{4} x - 2 {cos}^{2} x + {cos}^{4} x = 1 - 2 {cos}^{2} x + 2 {cos}^{4} x \end{matrix}$

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1 + {cos}^{2} 2 x = 2 ({cos}^{4} x + {sin}^{4} x)

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