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

Answer

Expert–verified

Hide Steps

Answer

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

Steps

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 $\frac{1}{\sqrt{5}}$ .