Questions: Find the exact value or state that it is undefined. In the latter case, enter "DNE". tan(tan^(-1)(1))

Transcript text: Find the exact value or state that it is undefined. In the latter case, enter "DNE". \[ \tan \left(\tan ^{-1}(1)\right) \]

Solution

Solution Steps

To solve the given problem, we need to understand the relationship between the tangent function and its inverse. The inverse tangent function, \(\tan^{-1}(x)\), returns the angle whose tangent is \(x\). Therefore, \(\tan(\tan^{-1}(1))\) should return the value 1, since \(\tan^{-1}(1)\) is the angle whose tangent is 1.

Step 1: Understanding the Inverse Tangent

We start with the expression \( \tan(\tan^{-1}(1)) \). The function \( \tan^{-1}(1) \) gives us the angle whose tangent is 1. This angle is \( \frac{\pi}{4} \) radians.

Step 2: Evaluating the Tangent

Next, we evaluate \( \tan\left(\frac{\pi}{4}\right) \). The tangent of \( \frac{\pi}{4} \) is known to be 1. Therefore, we have: \[ \tan(\tan^{-1}(1)) = \tan\left(\frac{\pi}{4}\right) = 1 \]

Step 3: Numerical Approximation

The numerical evaluation of \( \tan(\tan^{-1}(1)) \) yields approximately \( 0.9999999999999999 \). However, this value is very close to 1, and for practical purposes, we can round it to four significant digits.