The value of `sin ^(-1) (-(1)/(sqrt2)) + cos ^(-1) (-(1)/(2)) - tan ^(-1) (-sqrt3) + cot ^(-1) (-(1)/(sqrt3)) ` is

Options

A. `(5pi)/(12)`
B. `(17pi)/(12)`
C. 0
D. 1

Answer

To solve the expression \( \sin^{-1} \left(-\frac{1}{\sqrt{2}}\right) + \cos^{-1} \left(-\frac{1}{2}\right) - \tan^{-1} \left(-\sqrt{3}\right) + \cot^{-1} \left(-\frac{1}{\sqrt{3}}\right) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \sin^{-1} \left(-\frac{1}{\sqrt{2}}\right) \)

The value of \( \sin^{-1} x \) is defined in the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

Since \( \sin^{-1} \left(-\frac{1}{\sqrt{2}}\right) = -\sin^{-1} \left(\frac{1}{\sqrt{2}}\right) \), we know that:

\[ \sin^{-1} \left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \]

Thus,

\[ \sin^{-1} \left(-\frac{1}{\sqrt{2}}\right) = -\frac{\pi}{4} \]

### Step 2: Evaluate \( \cos^{-1} \left(-\frac{1}{2}\right) \)

The value of \( \cos^{-1} x \) is defined in the range \( [0, \pi] \).

We know that:

\[ \cos \theta = -\frac{1}{2} \implies \theta = \frac{2\pi}{3} \]

Thus,

\[ \cos^{-1} \left(-\frac{1}{2}\right) = \frac{2\pi}{3} \]

### Step 3: Evaluate \( -\tan^{-1} \left(-\sqrt{3}\right) \)

The value of \( \tan^{-1} x \) is defined in the range \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

Since \( -\tan^{-1} \left(-\sqrt{3}\right) = \tan^{-1} \left(\sqrt{3}\right) \), we know that:

\[ \tan \theta = \sqrt{3} \implies \theta = \frac{\pi}{3} \]

Thus,

\[ -\tan^{-1} \left(-\sqrt{3}\right) = \frac{\pi}{3} \]

### Step 4: Evaluate \( \cot^{-1} \left(-\frac{1}{\sqrt{3}}\right) \)

The value of \( \cot^{-1} x \) is defined in the range \( (0, \pi) \).

Since \( \cot^{-1} \left(-\frac{1}{\sqrt{3}}\right) = \pi - \cot^{-1} \left(\frac{1}{\sqrt{3}}\right) \), we know that:

\[ \cot \theta = \frac{1}{\sqrt{3}} \implies \theta = \frac{\pi}{6} \]

Thus,

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

### Step 5: Combine all the evaluated terms

Now we can combine all the evaluated terms:

\[ \sin^{-1} \left(-\frac{1}{\sqrt{2}}\right) + \cos^{-1} \left(-\frac{1}{2}\right) - \tan^{-1} \left(-\sqrt{3}\right) + \cot^{-1} \left(-\frac{1}{\sqrt{3}}\right) \]

Substituting the values we found:

\[ -\frac{\pi}{4} + \frac{2\pi}{3} + \frac{\pi}{3} + \frac{5\pi}{6} \]

### Step 6: Find a common denominator and simplify

The common denominator for \( 4, 3, 3, \) and \( 6 \) is \( 12 \).

Converting each term:

- \( -\frac{\pi}{4} = -\frac{3\pi}{12} \)
- \( \frac{2\pi}{3} = \frac{8\pi}{12} \)
- \( \frac{\pi}{3} = \frac{4\pi}{12} \)
- \( \frac{5\pi}{6} = \frac{10\pi}{12} \)

Now, adding these fractions:

\[ -\frac{3\pi}{12} + \frac{8\pi}{12} + \frac{4\pi}{12} + \frac{10\pi}{12} = \frac{19\pi}{12} \]

### Final Answer

Thus, the value of the expression is:

\[ \frac{19\pi}{12} \]