cos^(4)alpha-sin^(4)alpha=cos 2alpha

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Recognize identity: Recognize the identity to be used. We can use the Pythagorean identity sin^2(alpha) + cos^2(alpha) = 1 and the double angle formula for cosine, which is cos(2alpha) = cos^2(alpha) - sin^2(alpha).Express in squared terms: Express cos^(4)alpha - sin^(4)alpha in terms of squared terms. We can rewrite the expression as (cos^2(alpha))^2 - (sin^2(alpha))^2.Apply difference of squares: Apply the difference of squares formula. The difference of squares formula is a^2 - b^2 = (a + b)(a - b). Applying this to our expression, we get (cos^2(alpha) + sin^2(alpha))(cos^2(alpha) - sin^2(alpha)).Substitute Pythagorean identity: Substitute the Pythagorean identity into the expression. Since sin^2(alpha) + cos^2(alpha) = 1, we can replace the first part of the product with 1, resulting in 1 * (cos^2(alpha) - sin^2(alpha)).Simplify the expression: Simplify the expression. Simplifying the expression, we get cos^2(alpha) - sin^2(alpha).Recognize double angle formula: Recognize the double angle formula for cosine. The double angle formula for cosine is cos(2alpha) = cos^2(alpha) - sin^2(alpha).Compare with double angle formula: Compare the simplified expression with the double angle formula. We see that cos^2(alpha) - sin^2(alpha) is indeed the double angle formula for cosine, which is cos(2alpha).Conclude the original expression: Conclude that the original expression is equal to the double angle formula. Therefore, cos^(4)alpha - sin^(4)alpha is equal to cos 2alpha.

$\cos ^{4} \alpha-\sin ^{4} \alpha=\cos 2 \alpha$

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cos⁡4α−sin⁡4α=cos⁡2α \cos ^{4} \alpha-\sin ^{4} \alpha=\cos 2 \alpha cos4α−sin4α=cos2α

$\cos ^{4} \alpha-\sin ^{4} \alpha=\cos 2 \alpha$