To rewrite the trigonometric expression as an algebraic expression in x, we can make use of the relationship between cosine and sine functions.
The expression sin(cos^-1 x) represents the sine of the inverse cosine of x. Since the range of the inverse cosine function is limited to [0, π], and the sine function operates within that domain, we can use the following identity:
sin(cos^-1 x) = √(1 - x^2)
This identity is derived from the Pythagorean identity sin^2θ + cos^2θ = 1, where θ is the angle resulting from the inverse cosine operation.
Therefore, the algebraic expression in x, free of trigonometric or inverse trigonometric functions, is:
√(1 - x^2) for -1 ≤ x ≤ 1