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The Pythagorean identity is a fundamental principle in trigonometry that connects the squares of sine and cosine functions to 1. The identity states:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This relationship is derived from the Pythagorean theorem applied to a right-angled triangle, where the hypotenuse equals 1. With this identity, one can easily express either sine or cosine in terms of the other. For example, rearranging the identity gives:

• \( \sin^2 \theta = 1 - \cos^2 \theta \)
• \( \cos^2 \theta = 1 - \sin^2 \theta \)

Here, the problem uses the identity to transform \( \cos^2(\arcsin x) \) into \( 1 - \sin^2(\arcsin x) \). Since \( \sin(\arcsin x) = x \), this results in \( 1 - x^2 \). Thus, the identity provides the means to simplify the equation and solve it effectively.

Trigonometric functions are essential for describing relationships between the angles and sides of triangles. In this context, the functions \( \sin t \) and \( \cos t \) are used in the parametric equations given for \( x(t) \) and \( y(t) \).

• Sine Function: Given as \( x(t) = \sin t \), it captures the vertical component of the angle \( t \).
• Cosine Function: Found in \( y(t) = 1 + \cos^2 t \), it denotes the horizontal component of \( t \,\) squaring it to measure squared distance along the horizontal.

The inverse function, \( \arcsin x \), is used to express \( t \) in terms of \( x \). This inverse operation is critical for substituting \( t \) from one parametric function into another, facilitating the derivation of non-parametric equations.

Algebraic manipulation is the process of rearranging and simplifying expressions to solve for a variable or to simplify an expression. In this task, it involves substituting functions and employing identities effectively.Initially, \( x(t) = \sin t \) was transformed into \( t = \arcsin x \). This allowed the substitution of \( t \) into the second equation, \( y(t) = 1 + \cos^2 t \). This manipulation forms the basis of connecting parametric equations to a more familiar non-parametric form relating \( x \) and \( y \).Once substituted, further algebraic manipulation applies the Pythagorean identity: \( y = 1 + \cos^2(\arcsin x) \) simplifies to \( y = 2 - x^2 \) after substituting \( \cos^2(\arcsin x) = 1 - x^2 \). These steps highlight the power of algebraic techniques in solving and simplifying complex mathematical problems.