Prove the identity. cos^(4)x-sin^(4)x=cos 2x

Apply Pythagorean identity: We will start by using the Pythagorean identity sin^2(x) + cos^2(x) = 1 to rewrite the left side of the equation in terms of cosines only. cos^(4)x - sin^(4)x = (cos^2(x))^2 - (sin^2(x))^2 We can express sin^2(x) as 1 - cos^2(x) to get: (cos^2(x))^2 - (1 - cos^2(x))^2Rewrite in terms of cosines: Now we will expand the squared terms: (cos^2(x))^2 - (1 - cos^2(x))^2 = cos^4(x) - (1 - 2cos^2(x) + cos^4(x))Expand and simplify: Next, we simplify the expression by combining like terms: cos^4(x) - 1 + 2cos^2(x) - cos^4(x) = 2cos^2(x) - 1Use double angle formula: We recognize that 2cos^2(x) - 1 is the double angle formula for cosine: 2cos^2(x) - 1 = cos(2x)Final result: We have shown that the left side of the equation simplifies to the right side: cos^(4)x - sin^(4)x = cos(2x)

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Prove the identity.

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

$76$ . Prove the identity. $\newline$ $\cos ^{4} x-\sin ^{4} x=\cos 2 x$

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76

. Prove the identity.

\newline

\cos ^{4} x-\sin ^{4} x=\cos 2 x

Apply Pythagorean identity: We will start by using the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ to rewrite the left side of the equation in terms of cosines only. $\newline$ $\cos^{4}x - \sin^{4}x = (\cos^2(x))^2 - (\sin^2(x))^2$ $\newline$ We can express $\sin^2(x)$ as $1 - \cos^2(x)$ to get: $\newline$ $(\cos^2(x))^2 - (1 - \cos^2(x))^2$
Rewrite in terms of cosines: Now we will expand the squared terms: $\newline$ $(\cos^2(x))^2 - (1 - \cos^2(x))^2 = \cos^4(x) - (1 - 2\cos^2(x) + \cos^4(x))$
Expand and simplify: Next, we simplify the expression by combining like terms: $\cos^4(x) - 1 + 2\cos^2(x) - \cos^4(x) = 2\cos^2(x) - 1$
Use double angle formula: We recognize that $2\cos^2(x) - 1$ is the double angle formula for cosine: $\newline$ $2\cos^2(x) - 1 = \cos(2x)$
Final result: We have shown that the left side of the equation simplifies to the right side: $\cos^{4}x - \sin^{4}x = \cos(2x)$