Give the exact value of the expression without using a calculator. \[ \cos \left(\tan ^{-1}(-2)\right) \] \[ \cos \left(\tan ^{-1}(-2)\right)= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Step 1 ：The expression is asking for the cosine of the angle whose tangent is -2.

Step 2 ：We can use the identity \(\cos(\tan^{-1}(x)) = \frac{1}{\sqrt{1+x^2}}\) to simplify the expression.

Step 3 ：Substitute x = -2 into the expression, we get \(\cos(\tan^{-1}(-2)) = \frac{1}{\sqrt{1+(-2)^2}}\).

Step 4 ：Simplify the expression, we get \(\cos(\tan^{-1}(-2)) = \frac{1}{\sqrt{5}}\).

Step 5 ：Final Answer: The exact value of the expression \(\cos (\tan ^{-1}(-2))\) is \(\boxed{\frac{1}{\sqrt{5}}}\).