Use a right triangle to write the following expression as an algebraic expression. Assume that $\mathrm{x}$ is positive and that the given inverse trigonometric function is defined for the expression in $x$
\[
\cos \left(\sin ^{-1} 2 x\right)
\]

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 \(\boxed{\sqrt{1 - 4x^2}}\).