cos 6x=32cos^(6)x-48cos^(4)x+18cos^(2)x-1

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Recognize Equation Pattern: Given the equation cos 6x = 32cos^6(x) - 48cos^4(x) + 18cos^2(x) - 1, we recognize that the right side of the equation resembles the expanded form of a binomial raised to a power. Specifically, it looks like the expansion of (cos x - 1)^6 using the binomial theorem. Let's check if this is the case by expanding (cos x - 1)^6 and comparing it to the given polynomial.Expand Binomial Theorem: First, we need to expand (cos x - 1)^6 using the binomial theorem. The binomial theorem states that (a - b)^n = Σ (n choose k) * a^(n-k) * (-b)^k, where Σ denotes the sum over k from 0 to n. However, this expansion is quite lengthy and prone to errors, so we will look for a pattern or a simpler way to approach the problem.Identify Chebyshev Polynomial: Upon closer inspection, we realize that the given polynomial is actually a form of a Chebyshev polynomial of the first kind, which is defined by T_n(cos θ) = cos(nθ). The polynomial on the right side of the equation is T_6(cos x) because the highest power of cos is 6, which corresponds to the 6 in cos 6x on the left side of the equation. Therefore, we can rewrite the equation as cos 6x = T_6(cos x).Simplify Equation: Knowing that T_6(cos x) = cos 6x, we can simplify the original equation to cos 6x = cos 6x. This means that the equation is true for all values of x, as both sides are identical.Determine Solutions: Since the equation is true for all values of x, we can say that the solutions to the equation are all real numbers x.

$\cos 6x = 32\cos^6 x - 48\cos^4 x + 18\cos^2 x - 1$

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cos⁡6x=32cos⁡6x−48cos⁡4x+18cos⁡2x−1\cos 6x = 32\cos^6 x - 48\cos^4 x + 18\cos^2 x - 1cos6x=32cos6x−48cos4x+18cos2x−1

$\cos 6x = 32\cos^6 x - 48\cos^4 x + 18\cos^2 x - 1$