Find the exact value of the expression. \[ \cos \left[\sin ^{-1}\left(\frac{\sqrt{2}}{2}\right)\right] \]

Step 1 ：The problem is asking for the cosine of the arcsine of a value. The arcsine of a value is the angle whose sine is that value. So, we first need to find the angle whose sine is \(\frac{\sqrt{2}}{2}\). Then, we need to find the cosine of that angle.

Step 2 ：The sine of an angle in a right triangle is defined as the length of the opposite side divided by the length of the hypotenuse. So, we are looking for an angle in a right triangle where the length of the opposite side is \(\sqrt{2}\) and the length of the hypotenuse is 2. This is the case in a 45-45-90 triangle, where the angles are 45 degrees, 45 degrees, and 90 degrees, and the sides are in the ratio 1:1:\(\sqrt{2}\). So, the angle we are looking for is 45 degrees or \(\frac{\pi}{4}\) radians.

Step 3 ：The cosine of an angle in a right triangle is defined as the length of the adjacent side divided by the length of the hypotenuse. In a 45-45-90 triangle, the length of the adjacent side is also \(\sqrt{2}\), so the cosine of the angle is also \(\frac{\sqrt{2}}{2}\).

Step 4 ：Due to the limitations of floating point precision, the result is not exactly \(\frac{\sqrt{2}}{2}\). To get the exact value, we can use the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\). Since \(\sin(x) = \frac{\sqrt{2}}{2}\), we can solve for \(\cos(x)\) to get \(\cos(x) = \sqrt{1 - \sin^2(x)} = \sqrt{1 - \left(\frac{\sqrt{2}}{2}\right)^2} = \sqrt{1 - \frac{1}{2}} = \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2}\).

Step 5 ：Final Answer: The exact value of the expression is \(\boxed{\frac{\sqrt{2}}{2}}\).