MHB Proving Identity: $\sin^8A-\cos^8A$

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The discussion focuses on proving the identity $\sin^8(A) - \cos^8(A)$ through various algebraic manipulations. It starts by expressing the left-hand side in terms of $\sin^2(A)$ and $\cos^2(A)$, utilizing the difference of squares formula. Participants explore factorizations and substitutions to simplify the expression, ultimately leading to the conclusion that the identity can be rewritten in terms of squares of sine and cosine. The key takeaway is the importance of reducing higher powers to squares for easier manipulation and proof. The discussion emphasizes the algebraic relationships between sine and cosine in trigonometric identities.
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$\sin^8\left({A}\right)-\cos^8\left({A}\right)=(\sin^2\left({A}\right)-\cos^2\left({A}\right)(1-2\sin^2\left({A}\right)\cos^2\left({A}\right))$

$L.H.S=(\sin^2\left({A}\right)-\cos^2\left({A}\right)(1-2\sin^2\left({A}\right)\cos^2\left({A}\right))$

$ =(\sin\left({A}\right)+\cos\left({A}\right)) (\sin\left({A}\right)-\cos\left({A}\right)(\cos^2\left({A}\right)-\sin^2\left({A}\right))^2$

Any ideas?
 
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Silver Bolt said:
$\sin^8\left({A}\right)-\cos^8\left({A}\right)=(\sin^2\left({A}\right)-\cos^2\left({A}\right)(1-2\sin^2\left({A}\right)\cos^2\left({A}\right))$

$L.H.S=(\sin^2\left({A}\right)-\cos^2\left({A}\right)(1-2\sin^2\left({A}\right)\cos^2\left({A}\right))$

$ =(\sin\left({A}\right)+\cos\left({A}\right)) (\sin\left({A}\right)-\cos\left({A}\right)(\cos^2\left({A}\right)-\sin^2\left({A}\right))^2$

Any ideas?

$\sin^8\left({A}\right)-\cos^8\left({A}\right)$
= $(\sin^4\left({A}\right)+\cos^4\left({A}\right))(\sin^4\left({A}\right)-\cos^4\left({A}\right)$ using $a^2-b^2= (a+b)(a-b)$
= $(\sin^4\left({A}\right)+\cos^4\left({A}\right))(\sin^2\left({A}\right)+\cos^2\left({A}\right)(\sin^2\left({A}\right)-
\cos^2\left({A}\right))$ using $a^2-b^2= (a+b)(a-b)$
= $(\sin^4\left({A}\right)+\cos^4\left({A}\right))(\sin^2\left({A}\right)-
\cos^2\left({A}\right))$
= $((\sin^2\left({A}\right)+\cos^2\left({A}\right))^2-2 \sin^2\left({A}\right)\cos^2\left({A}\right)(\sin^2\left({A}\right)-
\cos^2\left({A}\right))$ using $a^2+b^2 = + (a+b)^2 - 2ab$
= $((\sin^2\left({A}\right)+\cos^2\left({A}\right))^2-2 \sin^2\left({A}\right)\cos^2\left({A}\right)(\sin^2\left({A}\right)-
\cos^2\left({A}\right))$
= $(1-2 \sin^2\left({A}\right)\cos^2\left({A}\right)(\sin^2\left({A}\right)-
\cos^2\left({A}\right))$

The rational is that we need to reduce the power to squares of sin and cos
 
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