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The expression in the image involves nested trigonometric functions. It appears as follows:

\[
\sin \left[ \tan^{-1} \left( \frac{2x}{1 - x^2} \right) + \tan^{-1} \left( \frac{8x}{1 - 16x^2} \right) + \tan^{-1} \left( \frac{3x}{1 - 9x^2} \right) \right]
\]

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Here are some related questions to explore:

1. What is the formula for simplifying sums of inverse tangent (arctan) functions?
2. How can we simplify expressions involving nested trigonometric identities?
3. What is the domain of this trigonometric function?
4. How does the sine function behave with sum of angles?
5. Can we use substitution to simplify this type of expression?

**Tip**: When dealing with sums of inverse tangent functions, you can use the identity:
\[
\tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \quad \text{if} \quad ab < 1.
\]

Solve the expression involving nested trigonometric functions: sin[arctan(2x / (1 - x^2)) + arctan(8x / (1 - 16x^2)) + arctan(3x / (1 - 9x^2))]

Learn how to simplify and solve the nested trigonometric expression involving the sine of sums of arctangent functions. This solution breaks down key concepts such as the sum of inverse tangents and the sine of sums of angles, suitable for high school trigonometry students in Grades 11-12.

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