Step 1 :The given expression is \(\cos(\sin^{-1}(2x))\). We know that \(\sin^{-1}(2x)\) represents an angle whose sine is \(2x\).
Step 2 :We can represent this in a right triangle where the opposite side is \(2x\) and the hypotenuse is \(1\).
Step 3 :By the Pythagorean theorem, the adjacent side would be \(\sqrt{1 - (2x)^2}\).
Step 4 :The cosine of an angle in a right triangle is given by the ratio of the adjacent side to the hypotenuse.
Step 5 :Therefore, \(\cos(\sin^{-1}(2x))\) can be represented as \(\frac{\sqrt{1 - (2x)^2}}{1}\), which simplifies to \(\sqrt{1 - (2x)^2}\).
Step 6 :Final Answer: The algebraic expression for \(\cos (\sin ^{-1} 2 x)\) is \(\boxed{\sqrt{1 - 4x^2}}\).