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Basic Trigonometric Identities

Prove 3 - ...

Question

Prove $3 - 4 cos 2 α + cos 4 α = 8 {sin}^{4} α .$

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Solution

$3 - 4 cos 2 α + cos 4 α$
$= 3 - 4 (1 - {2 sin}^{2} α) + (1 - {2 sin}^{2} 2 α)$
$= 3 - 4 + {8 sin}^{2} α + 1 - {8 sin}^{2} α {cos}^{2} α$
$= {8 sin}^{2} α (1 - {cos}^{2} α)$
$= {8 sin}^{4} α$

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