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The double angle identity is a powerful tool in trigonometry, especially when dealing with angles that are twice another angle. The specific formula for the sine of a double angle is given by:

• \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \)

This identity tells us that when we have an angle that is doubled, the sine of that angle can be expressed in terms of the sine and cosine of the original angle \(\theta\). It is particularly useful in breaking down complex trigonometric expressions into simpler components.
If you come across an expression like \( \sin(2\sin^{-1} x) \), you can rewrite it using this identity, by identifying \( \theta = \sin^{-1} x \). Here, \( \sin^{-1} x \) means the angle whose sine is \( x \).
Once you replace \( 2\theta \) with \( 2\sin^{-1} x \) in the identity, you get:

• \( \sin(2\sin^{-1} x) = 2 \sin(\sin^{-1} x) \cos(\sin^{-1} x) \)

This expression can further be simplified with the help of another trigonometric identity and the properties of inverse trigonometry.

Inverse trigonometric functions, such as \( \sin^{-1} x \), \( \cos^{-1} x \), and \( \tan^{-1} x \), allow us to find an angle when we know its trigonometric value. For example, when we know \( x \) as a sine value, \( \sin^{-1} x \) gives us the angle whose sine is \( x \).

• The function \( \sin^{-1} x \) is specifically used to find the angle \( \theta \) for \( -1 \leq x \leq 1 \), where \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \).

Understanding inverse trigonometric functions is crucial because it allows for the translation of trigonometric values into angles. This conversion can be used in conjunction with other identities to solve equations or simplify expressions.
For instance, in our problem, \( \sin^{-1} x \) is the angle whose sine is x. Hence, \( \sin(\sin^{-1} x) = x \). But we also need to find \( \cos(\sin^{-1} x) \). Here the Pythagorean identity helps us again: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity allows us to express the cosine in terms of sine:

• \( \cos(\sin^{-1} x) = \sqrt{1 - x^2} \)

This expression for \( \cos(\sin^{-1} x) \) is vital for simplifying the trigonometric expression.

Simplifying trigonometric expressions involves rewriting them in a simpler or more convenient form. This process often uses various trigonometric identities and properties. In our exercise, we're aiming to simplify the expression \( \sin(2\sin^{-1} x) \).
We've already seen that using the double angle identity, we get:

• \( \sin(2\sin^{-1} x) = 2 \sin(\sin^{-1} x) \cos(\sin^{-1} x) \)
• Substituting known values gives: \( \sin(\sin^{-1} x) = x \) and \( \cos(\sin^{-1} x) = \sqrt{1-x^2} \)

Plug these expressions back into the formula:

• \( \sin(2\sin^{-1} x) = 2x\sqrt{1-x^2} \)

This final form of the expression is simpler and in terms of a single variable \( x \), providing a clearer view of how the trigonometric relationships work.
By breaking down the components step-by-step—using the double angle identity, understanding inverse trigonometric functions, and applying algebraic identities—you can not only simplify expressions effectively but also deepen your understanding of trigonometric principles.