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Solve for [tex]\( x \)[/tex].

[tex]\[ 3x = 6x - 2 \][/tex]

---

[tex]\[
\begin{aligned}
\cos^8 A - \sin^8 A & = (\cos^2 A - \sin^2 A)(1 - 2 \sin^2 A \cos^2 A)
\end{aligned}
\][/tex]

[tex]\[
\sin^8 A + \cos^8 A = 1 - 4 \sin^2 A \cos^2 A + 2 \sin^4 A \cos^4 A
\][/tex]



Answer :

GinnyAnswer
Let's start by exploring the given equations and the expected results from the solution process.

1. Simplifying [tex]\(\cos^8A - \sin^8A\)[/tex]:

First, we factorize the left-hand side [tex]\(\cos^8A - \sin^8A\)[/tex] using algebraic identities.

Step 1: Recognize the difference of powers.
[tex]\[ \cos^8A - \sin^8A = (\cos^4A)^2 - (\sin^4A)^2 \][/tex]

Step 2: Apply the difference of squares identity, [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
[tex]\[ (\cos^4A)^2 - (\sin^4A)^2 = (\cos^4A - \sin^4A)(\cos^4A + \sin^4A) \][/tex]

Step 3: Further break down [tex]\(\cos^4A - \sin^4A\)[/tex] using the same identity.
[tex]\[ \cos^4A - \sin^4A = (\cos^2A)^2 - (\sin^2A)^2 = (\cos^2A - \sin^2A)(\cos^2A + \sin^2A) \][/tex]

Step 4: Simplify and combine all factors.
[tex]\[ \cos^8A - \sin^8A = (\cos^2A - \sin^2A)(\cos^2A + \sin^2A)(\cos^4A + \sin^4A) \][/tex]

Since [tex]\(\cos^2A + \sin^2A = 1\)[/tex], the equation simplifies to:
[tex]\[ \cos^8A - \sin^8A = (\cos^2A - \sin^2A)(1)(\cos^4A + \sin^4A) \][/tex]
[tex]\[ \cos^8A - \sin^8A = (\cos^2A - \sin^2A)(\cos^4A + \sin^4A) \][/tex]

2. Simplifying [tex]\(\sin^8A + \cos^8A\)[/tex]:

This summation requires us to combine the powers more straightforwardly.

Step 1: Notice that the combination [tex]\(\sin^8A + \cos^8A\)[/tex] is often approached without factorization because it’s a straightforward addition.

Step 2: Use trigonometric identities to simplify or verify the expression step-by-step.
Given the identity for [tex]\((\sin^2A + \cos^2A)^4 = 1\)[/tex], we know:
[tex]\[ \sin^8A + \cos^8A = (\sin^2A + \cos^2A)^4 - 4\sin^2A\cos^2A(\sin^2A + \cos^2A)^2 + \sin^4A\cos^4A \][/tex]
Since [tex]\(\sin^2A + \cos^2A = 1\)[/tex], this simplifies to:
[tex]\[ \sin^8A + \cos^8A = 1 - 4\sin^2A\cos^2A + 2\sin^4A\cos^4A \][/tex]

Therefore, we confirm the simplified results for both expressions:

1. [tex]\(\cos^8A - \sin^8A\)[/tex] is factored as:
[tex]\[ (\cos^2A - \sin^2A)(\cos^4A + \sin^4A) \][/tex]

2. [tex]\(\sin^8A + \cos^8A\)[/tex] simplifies to:
[tex]\[ 1 - 4\sin^2A\cos^2A + 2\sin^4A\cos^4A \][/tex]

The detailed step-by-step breakdown helps us reach these conclusions while aligning with known trigonometric identities and algebraic properties.

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