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The equation in the image is:

\[
\frac{\sqrt{x + 8} \cdot \ln{(x^2 - 1)} - \left(\frac{3x^2 - \frac{2x - 8}{x}}{x}\right)}{\cos^{-1}\left(\frac{2x - 3}{x + 1}\right) + \sin{\left(\frac{2\pi}{3x + 2}\right)}} = \frac{\pi}{3}
\]

Would you like a detailed solution or explanation on how to solve this equation? If so, please specify any part of the equation you'd like to focus on.

Here are 5 questions related to the equation:

1. What is the domain of the function \(\ln{(x^2 - 1)}\)?
2. How do you simplify the expression \(\frac{3x^2 - \frac{2x - 8}{x}}{x}\)?
3. What are the principal values for \(\cos^{-1}(x)\) and \(\sin(x)\)?
4. How would you approach solving this equation for \(x\)?
5. What techniques can be used to simplify the expression before solving?

**Tip:** When solving equations with logarithmic, trigonometric, and square root functions, it's crucial to consider the domain of each function to avoid invalid solutions.

Solve the equation involving square roots, logarithms, and trigonometric functions: \[\frac{\sqrt{x + 8} \cdot \ln{(x^2 - 1)} - \left(\frac{3x^2 - \frac{2x - 8}{x}}{x}\right)}{\cos^{-1}\left(\frac{2x - 3}{x + 1}\right) + \sin{\left(\frac{2\pi}{3x + 2}\right)}} = \frac{\pi}{3}\]

Explore a step-by-step solution for solving an advanced equation involving logarithmic, trigonometric, and inverse trigonometric functions. Understand the properties and domains of each function to find the valid solutions. Ideal for students in Grades 11-12 looking to deepen their understanding of complex mathematical concepts.

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