The value of
`cot^(-1)(-1)+cosec^(-1)(-sqrt(2))+sec^(-1)(2)` is

Options

A. `(5pi)/(6)`
B. `(2pi)/(3)`
C. `(7pi)/(6)`
D. `(pi)/(6)`

Explanation

To solve the expression \( \cot^{-1}(-1) + \csc^{-1}(-\sqrt{2}) + \sec^{-1}(2) \), we will evaluate each term step by step.

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

The cotangent function is negative in the second and fourth quadrants. The angle whose cotangent is -1 is \( \frac{3\pi}{4} \) (in the second quadrant).

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

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

We know that \( \csc^{-1}(x) \) is the angle whose cosecant is \( x \). Therefore, we need to find the angle \( \theta \) such that \( \csc(\theta) = -\sqrt{2} \).

This implies that \( \sin(\theta) = -\frac{1}{\sqrt{2}} \), which corresponds to \( \theta = -\frac{\pi}{4} \) (in the fourth quadrant).

Thus, we can express this as:
\[ \csc^{-1}(-\sqrt{2}) = -\frac{\pi}{4} \]

### Step 3: Evaluate \( \sec^{-1}(2) \)

The secant function is the reciprocal of the cosine function. Therefore, we need to find the angle \( \phi \) such that \( \sec(\phi) = 2 \).

This implies that \( \cos(\phi) = \frac{1}{2} \), which corresponds to \( \phi = \frac{\pi}{3} \) (in the first quadrant).

Thus, we have:
\[ \sec^{-1}(2) = \frac{\pi}{3} \]

### Step 4: Combine the results

Now we can combine all the evaluated results:
\[ \cot^{-1}(-1) + \csc^{-1}(-\sqrt{2}) + \sec^{-1}(2) = \frac{3\pi}{4} - \frac{\pi}{4} + \frac{\pi}{3} \]

### Step 5: Simplify the expression

First, combine \( \frac{3\pi}{4} - \frac{\pi}{4} \):
\[ \frac{3\pi}{4} - \frac{\pi}{4} = \frac{2\pi}{4} = \frac{\pi}{2} \]

Now, we need to add \( \frac{\pi}{2} + \frac{\pi}{3} \). To do this, we need a common denominator, which is 6:
\[ \frac{\pi}{2} = \frac{3\pi}{6}, \quad \frac{\pi}{3} = \frac{2\pi}{6} \]

Now add them:
\[ \frac{3\pi}{6} + \frac{2\pi}{6} = \frac{5\pi}{6} \]

### Final Answer
Thus, the value of \( \cot^{-1}(-1) + \csc^{-1}(-\sqrt{2}) + \sec^{-1}(2) \) is:
\[ \frac{5\pi}{6} \]

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