What is the value of the following functions (using principal value)
`tan^(-1)(1) - cot^(-1)(-1)`

Answer

To find the value of the expression \( \tan^{-1}(1) - \cot^{-1}(-1) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \tan^{-1}(1) \)

The function \( \tan^{-1}(x) \) gives the angle whose tangent is \( x \). We need to find the angle \( \theta \) such that:

\[ \tan(\theta) = 1 \]

The principal value of \( \tan^{-1}(1) \) is \( \frac{\pi}{4} \) because:

\[ \tan\left(\frac{\pi}{4}\right) = 1 \]

Thus,

\[ \tan^{-1}(1) = \frac{\pi}{4} \]

### Step 2: Evaluate \( \cot^{-1}(-1) \)

Next, we evaluate \( \cot^{-1}(-1) \). The function \( \cot^{-1}(x) \) gives the angle whose cotangent is \( x \). We need to find the angle \( \phi \) such that:

\[ \cot(\phi) = -1 \]

The cotangent is negative in the second and fourth quadrants. The angle where \( \cot(\phi) = -1 \) corresponds to:

\[ \phi = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]

Thus,

\[ \cot^{-1}(-1) = \frac{3\pi}{4} \]

### Step 3: Combine the results

Now, we can substitute these values back into the original expression:

\[ \tan^{-1}(1) - \cot^{-1}(-1) = \frac{\pi}{4} - \frac{3\pi}{4} \]

Calculating this gives:

\[ \frac{\pi}{4} - \frac{3\pi}{4} = -\frac{2\pi}{4} = -\frac{\pi}{2} \]

### Final Answer

Thus, the value of the expression \( \tan^{-1}(1) - \cot^{-1}(-1) \) is:

\[ -\frac{\pi}{2} \]

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