The value of `sec(sin^-1(sin((-50pi)/9))+cos^-1(cos(31pi)/9))`

Options

A. `sec.(10pi)/(9)`
B. `sec.(pi)/(9)`
C. 1
D. `-1`

Explanation

To solve the expression \( \sec(\sin^{-1}(\sin(-\frac{50\pi}{9})) + \cos^{-1}(\cos(\frac{31\pi}{9}))) \), we will break it down into manageable parts.

### Step 1: Simplify \( \sin^{-1}(\sin(-\frac{50\pi}{9})) \)

First, we need to simplify \( \sin^{-1}(\sin(-\frac{50\pi}{9})) \).

1. **Convert the angle**:
\[ -\frac{50\pi}{9} = -6\pi + \frac{4\pi}{9} \]
This is because \( -6\pi \) is a multiple of \( 2\pi \) and we can add \( 6\pi \) to the angle to bring it within the principal range of the sine function.

2. **Use the property of sine**:
\[ \sin(-\frac{50\pi}{9}) = \sin(4\pi/9) \]
Thus,
\[ \sin^{-1}(\sin(-\frac{50\pi}{9})) = \sin^{-1}(\sin(4\pi/9)) = \frac{4\pi}{9} \]

### Step 2: Simplify \( \cos^{-1}(\cos(\frac{31\pi}{9})) \)

Next, we simplify \( \cos^{-1}(\cos(\frac{31\pi}{9})) \).

1. **Convert the angle**:
\[ \frac{31\pi}{9} = 4\pi - \frac{5\pi}{9} \]
This is because \( 4\pi \) is a multiple of \( 2\pi \) and we can subtract \( 4\pi \) to bring it within the principal range of the cosine function.

2. **Use the property of cosine**:
\[ \cos(\frac{31\pi}{9}) = \cos(4\pi - \frac{5\pi}{9}) = \cos(\frac{5\pi}{9}) \]
Thus,
\[ \cos^{-1}(\cos(\frac{31\pi}{9})) = \cos^{-1}(\cos(\frac{5\pi}{9})) = \frac{5\pi}{9} \]

### Step 3: Combine the results

Now we combine the results from Steps 1 and 2:
\[ \sin^{-1}(\sin(-\frac{50\pi}{9})) + \cos^{-1}(\cos(\frac{31\pi}{9})) = \frac{4\pi}{9} + \frac{5\pi}{9} = \pi \]

### Step 4: Find \( \sec(\pi) \)

Finally, we need to find:
\[ \sec(\pi) = \frac{1}{\cos(\pi)} = \frac{1}{-1} = -1 \]

### Final Answer

Thus, the value of the expression is:
\[ \sec(\sin^{-1}(\sin(-\frac{50\pi}{9})) + \cos^{-1}(\cos(\frac{31\pi}{9}))) = -1 \]

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