To simplify the expression $\tan(\sin^{-1}(x))$, follow these steps:

Step 1: Understand the given expression

The expression involves the inverse sine function ($\sin^{-1}(x)$) and the tangent function ($\tan$).

Let $\theta = \sin^{-1}(x)$. This means that: $\sin(\theta) = x$ Thus, $\theta$ is the angle whose sine is $x$.

Step 2: Apply a right triangle interpretation

Consider a right triangle where: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{1}$ This implies that the opposite side is $x$ and the hypotenuse is 1. Using the Pythagorean theorem, we can find the adjacent side: $\text{adjacent} = \sqrt{1^2 - x^2} = \sqrt{1 - x^2}$

Step 3: Use the definition of $\tan(\theta)$

For the same angle $\theta$, the tangent is: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{\sqrt{1 - x^2}}$

Final Answer:

$\tan(\sin^{-1}(x)) = \frac{x}{\sqrt{1 - x^2}}$ This is the simplified form of the given expression.

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Would you like more details or have any further questions? Here are some related questions:

1. How do you simplify $\tan(\cos^{-1}(x))$?
2. What is the simplified form of $\sin(\tan^{-1}(x))$?
3. Can you derive $\tan(\sin^{-1}(x))$ using trigonometric identities?
4. What is the domain of the expression $\tan(\sin^{-1}(x))$?
5. How do you find $\cot(\sin^{-1}(x))$?

Tip: Always check the domain of the function when dealing with inverse trigonometric expressions, as they impose restrictions on the variable $x$.