Find the principal values of the following :
`tan^(-1) sqrt(3) - sec^(-1) (-2)` is equal to :

Options

A. `pi`
B. `-pi/3`
C. `pi/3`
D. `(2pi)/3`

Answer

To find the principal value of the expression \( \tan^{-1}(\sqrt{3}) - \sec^{-1}(-2) \), we will follow these steps:

### Step 1: Calculate \( \tan^{-1}(\sqrt{3}) \)
The value of \( \tan^{-1}(\sqrt{3}) \) corresponds to the angle whose tangent is \( \sqrt{3} \). This occurs at:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]
Thus,
\[ \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \]

### Step 2: Calculate \( \sec^{-1}(-2) \)
The secant function is defined as \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Therefore, if \( \sec(\theta) = -2 \), then:
\[ \cos(\theta) = -\frac{1}{2} \]
The angle \( \theta \) for which \( \cos(\theta) = -\frac{1}{2} \) is:
\[ \theta = \frac{2\pi}{3} \quad \text{(in the range of } [0, \pi] \text{ for } \sec^{-1}) \]
Thus,
\[ \sec^{-1}(-2) = \frac{2\pi}{3} \]

### Step 3: Substitute the values into the expression
Now we substitute the values we found into the original expression:
\[ \tan^{-1}(\sqrt{3}) - \sec^{-1}(-2) = \frac{\pi}{3} - \frac{2\pi}{3} \]

### Step 4: Simplify the expression
Now we simplify:
\[ \frac{\pi}{3} - \frac{2\pi}{3} = -\frac{\pi}{3} \]

### Final Answer
Thus, the principal value of the expression \( \tan^{-1}(\sqrt{3}) - \sec^{-1}(-2) \) is:
\[ -\frac{\pi}{3} \]

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