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