What is the value of cos⁡(tan-1⁡(45))?

The correct option is 5/√41.

Let's understand how to evaluate the expression $\cos \left({\tan }^{-1}\right(\frac{4}{5}\left)\right)$ step-by-step.

First, let's represent the inner inverse trigonometric function as an angle.
Let $\theta ={\tan }^{-1}\left(\frac{4}{5}\right)$.

By definition of the inverse tangent function, this is equivalent to:
$\tan \left(\theta \right)=\frac{4}{5}$

Recall that in a right-angled triangle, the tangent of an angle $\theta$ is the ratio of the length of the opposite side to the length of the adjacent side:
$\tan \left(\theta \right)=\frac{\text{Opposite}}{\text{Adjacent}}=\frac{4}{5}$

Therefore, we can model this with a right-angled triangle where:
• Length of the opposite side = 4
• Length of the adjacent side = 5

Next, we use the Pythagorean theorem to find the length of the hypotenuse ($h$):
$h^2={\text{Opposite}}^2+{\text{Adjacent}}^2$
$h^2=4^2+5^2$
$h^2=16+25$
$h^2=41$
$h=\sqrt{41}$

Now, we need to find the value of the original expression, which is $\cos \left(\theta \right)$.
In a right-angled triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse:
$\cos \left(\theta \right)=\frac{\text{Adjacent}}{\text{Hypotenuse}}$

Substituting the values we have:
$\cos \left(\theta \right)=\frac{5}{\sqrt{41}}$

Thus, the value of $\cos \left({\tan }^{-1}\right(\frac{4}{5}\left)\right)$ is indeed $\frac{5}{\sqrt{41}}$.