`sin^(-1)(-(1/2))+cos^(-1)(-(1/2))+cot^(-1)(-sqrt3)+cosec^(-1)(sqrt2)+tan^(-1)(-1)+sec^(-1)(sqrt2)` equals

Options

A. `(9pi)/4`
B. `(19pi)/12`
C. `(3pi)/2`
D. `pi/2`

Answer

To solve the expression \( \sin^{-1}\left(-\frac{1}{2}\right) + \cos^{-1}\left(-\frac{1}{2}\right) + \cot^{-1}\left(-\sqrt{3}\right) + \csc^{-1}\left(\sqrt{2}\right) + \tan^{-1}\left(-1\right) + \sec^{-1}\left(\sqrt{2}\right) \), we will use some properties of inverse trigonometric functions.

### Step-by-step Solution:

1. **Evaluate \( \sin^{-1}\left(-\frac{1}{2}\right) \)**:
\[ \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \]
(Since \( \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2} \))

2. **Evaluate \( \cos^{-1}\left(-\frac{1}{2}\right) \)**:
\[ \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \]
(Since \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \))

3. **Evaluate \( \cot^{-1}\left(-\sqrt{3}\right) \)**:
\[ \cot^{-1}\left(-\sqrt{3}\right) = \pi - \cot^{-1}(\sqrt{3}) = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \]
(Since \( \cot\left(\frac{\pi}{6}\right) = \sqrt{3} \))

4. **Evaluate \( \csc^{-1}\left(\sqrt{2}\right) \)**:
\[ \csc^{-1}\left(\sqrt{2}\right) = \frac{\pi}{4} \]
(Since \( \csc\left(\frac{\pi}{4}\right) = \sqrt{2} \))

5. **Evaluate \( \tan^{-1}\left(-1\right) \)**:
\[ \tan^{-1}\left(-1\right) = -\frac{\pi}{4} \]
(Since \( \tan\left(-\frac{\pi}{4}\right) = -1 \))

6. **Evaluate \( \sec^{-1}\left(\sqrt{2}\right) \)**:
\[ \sec^{-1}\left(\sqrt{2}\right) = \frac{\pi}{4} \]
(Since \( \sec\left(\frac{\pi}{4}\right) = \sqrt{2} \))

### Combine all the results:
Now we can combine all the evaluated values:
\[ -\frac{\pi}{6} + \frac{2\pi}{3} + \frac{5\pi}{6} + \frac{\pi}{4} - \frac{\pi}{4} + \frac{\pi}{4} \]

### Simplifying the expression:
1. Combine \( -\frac{\pi}{6} + \frac{5\pi}{6} \):
\[ -\frac{\pi}{6} + \frac{5\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3} \]

2. Now add \( \frac{2\pi}{3} + \frac{2\pi}{3} \):
\[ \frac{2\pi}{3} + \frac{2\pi}{3} = \frac{4\pi}{3} \]

3. Finally, add \( \frac{4\pi}{3} + \frac{\pi}{4} - \frac{\pi}{4} \):
\[ \frac{4\pi}{3} + 0 = \frac{4\pi}{3} \]

### Final Result:
Thus, the final result is:
\[ \frac{4\pi}{3} \]