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Is my answer correct? why or why not

Simplify the following expression:
[ tan (arcsin (x)) ]
[ √(1+x²) / x ]
[ √(1-x²) ]
[ √(1-x²) / x ]

1 Answer

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Final answer:

To verify the simplification of tan(arcsin(x)) multiplied by a series of radical expressions involving x, we must use trigonometric identities and simplify each term step by step. The tan(arcsin(x)) part simplifies to x/√(1 - x²), and combining this with the rest of the expression will ultimately cancel out terms, ideally resulting in 1 if done correctly. If any other value was obtained in the simplification, it is likely incorrect due to an error in the simplification process.

Step-by-step explanation:

To check if your answer to the expression tan(arcsin(x)) ∗ (√(1+x²) / x) ∗ √(1-x²) ∗ (√(1-x²) / x) is correct, we need to simplify the expression step by step. First, let's use the definition of the tangent function in terms of sine and cosine: tan(θ) = sin(θ) / cos(θ). Knowing that arcsin(x) is the angle whose sine is x, we can represent the cosine of this angle using the Pythagorean identity: cos(θ) = √(1 - sin²(θ)), which, in this case, becomes cos(arcsin(x)) = √(1 - x²).

The first part of the expression simplifies to: tan(arcsin(x)) = x / √(1 - x²). The remaining parts involve multiplying and dividing by x's and radical expressions. Simplifying further, the x's and the √(1 - x²) terms cancel out, leaving us with the simplified form, which should mathematically equal 1 if the initial expression is valid for the domain of x where arcsin and the square root functions are defined.

If your simplification resulted in a different answer, check each step for errors in algebraic manipulation or consider restrictions of the domain for the functions involved.

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