Rewrite sin left(2 cos ^{-1} wright) as an algebraic expression in w.

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Step:1 ：We know that (sin(cos^{-1} w)) and (cos(cos^{-1} w)) can be expressed in terms of (w). Specifically, (cos(cos^{-1} w) = w) and (sin(cos^{-1} w) = sqrt{1 - w^2}), because (sin^2(theta) + cos^2(theta) = 1) for any angle (theta).Step:2 ：Substituting these expressions into the formula gives us (2sin(cos ^{-1} w)cos(cos ^{-1} w) = 2sqrt{1 - w^2} cdot w).Step:3 ：Final Answer: (sin left(2 cos ^{-1} wright) = boxed{2wsqrt{1 - w^2}})

Answer

Step: 1 ：We know that (sin(cos^{-1} w)) and (cos(cos^{-1} w)) can be expressed in terms of (w). Specifically, (cos(cos^{-1} w) = w) and (sin(cos^{-1} w) = sqrt{1 - w^2}), because (sin^2(theta) + cos^2(theta) = 1) for any angle (theta).Step: 2 ：Substituting these expressions into the formula gives us (2sin(cos ^{-1} w)cos(cos ^{-1} w) = 2sqrt{1 - w^2} cdot w).Step: 3 ：Final Answer: (sin left(2 cos ^{-1} wright) = boxed{2wsqrt{1 - w^2}})

Rewrite $\sin \left(2 \cos ^{-1} w\right)$ as an algebraic expression in $w$.

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