Question

Find the exact value of$\cos \left({\tan }^{-1}2\right)$.

Short answer

The required value is$\frac{\sqrt{5}}{5}$.

Step-by-step solution

Step 1. Given Information

We need to find the value of the trigonometric function$\cos \left({\tan }^{-1}2\right)$.

Step 2. Finding the value

Let $$\begin{gathered} \theta ={\tan }^{-1}2 \\ \Rightarrow \tan \theta =2 \end{gathered}$$

The interval of tangent function is $0<\theta <\frac{\mathrm{\pi }}{2}$since $\tan \theta >0$.

The hypotenuse is $\sqrt{2^2+1^2}=\sqrt{5}$.

$$\begin{gathered} \therefore \cos \theta =\frac{1}{\sqrt{5}} \\ \Rightarrow \cos \theta =\frac{\sqrt{5}}{\sqrt{5}\times \sqrt{5}} \\ \Rightarrow \cos \theta =\frac{\sqrt{5}}{5} \end{gathered}$$