`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 \( \sin^{-1}(-\frac{1}{2}) + \cos^{-1}(-\frac{1}{2}) + \cot^{-1}(-\sqrt{3}) + \csc^{-1}(\sqrt{2}) + \tan^{-1}(-1) + \sec^{-1}(\sqrt{2}) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \sin^{-1}(-\frac{1}{2}) \)
Using the property \( \sin^{-1}(-x) = -\sin^{-1}(x) \):
\[ \sin^{-1}(-\frac{1}{2}) = -\sin^{-1}(\frac{1}{2}) = -\frac{\pi}{6} \]

**Hint:** Remember that the inverse sine function is odd.

### Step 2: Evaluate \( \cos^{-1}(-\frac{1}{2}) \)
Using the property \( \cos^{-1}(-x) = \pi - \cos^{-1}(x) \):
\[ \cos^{-1}(-\frac{1}{2}) = \pi - \cos^{-1}(\frac{1}{2}) = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

**Hint:** The inverse cosine function reflects across \( \pi/2 \).

### Step 3: Evaluate \( \cot^{-1}(-\sqrt{3}) \)
Using the property \( \cot^{-1}(-x) = \pi - \cot^{-1}(x) \):
\[ \cot^{-1}(-\sqrt{3}) = \pi - \cot^{-1}(\sqrt{3}) = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \]

**Hint:** The inverse cotangent function also reflects across \( \pi/2 \).

### Step 4: Evaluate \( \csc^{-1}(\sqrt{2}) \)
Since \( \csc^{-1}(x) \) is defined for positive \( x \):
\[ \csc^{-1}(\sqrt{2}) = \frac{\pi}{4} \]

**Hint:** The cosecant function is the reciprocal of sine.

### Step 5: Evaluate \( \tan^{-1}(-1) \)
Using the property \( \tan^{-1}(-x) = -\tan^{-1}(x) \):
\[ \tan^{-1}(-1) = -\tan^{-1}(1) = -\frac{\pi}{4} \]

**Hint:** The inverse tangent function is odd.

### Step 6: Evaluate \( \sec^{-1}(\sqrt{2}) \)
Since \( \sec^{-1}(x) \) is defined for positive \( x \):
\[ \sec^{-1}(\sqrt{2}) = \frac{\pi}{4} \]

**Hint:** The secant function is the reciprocal of cosine.

### Final Calculation
Now we combine all the results:
\[ -\frac{\pi}{6} + \frac{2\pi}{3} + \frac{5\pi}{6} + \frac{\pi}{4} - \frac{\pi}{4} + \frac{\pi}{4} \]

Combining the terms:
1. Combine \( -\frac{\pi}{6} + \frac{5\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3} \)
2. Now add \( \frac{2\pi}{3} + \frac{2\pi}{3} = \frac{4\pi}{3} \)
3. Finally, adding \( \frac{4\pi}{3} + 0 = \frac{4\pi}{3} \)

### Conclusion
The final result is:
\[ \frac{4\pi}{3} \]

### Summary of Steps:
1. Evaluate \( \sin^{-1}(-\frac{1}{2}) \) to get \( -\frac{\pi}{6} \).
2. Evaluate \( \cos^{-1}(-\frac{1}{2}) \) to get \( \frac{2\pi}{3} \).
3. Evaluate \( \cot^{-1}(-\sqrt{3}) \) to get \( \frac{5\pi}{6} \).
4. Evaluate \( \csc^{-1}(\sqrt{2}) \) to get \( \frac{\pi}{4} \).
5. Evaluate \( \tan^{-1}(-1) \) to get \( -\frac{\pi}{4} \).
6. Evaluate \( \sec^{-1}(\sqrt{2}) \) to get \( \frac{\pi}{4} \).
7. Combine all results to find the final answer.

**Final Answer:** \( \frac{4\pi}{3} \)