The value of the expression `sin^(-1)(sin(22pi)/7)cos^(-1)(cos(5pi)/3)+tan^(-1)(tan(5pi)/7)+sin^(-1)(cos2)` is `(17pi)/(42)-2` (b) `-2` `(-pi)/(21)-2` (d) `non eoft h e s e`

Options

A. `(17 pi)/(42) -2`
B. `-2`
C. `(-pi)/(21) -2`
D. none of these

Explanation

To solve the expression \( \sin^{-1}(\sin(22\pi/7)) + \cos^{-1}(\cos(5\pi/3)) + \tan^{-1}(\tan(5\pi/7)) + \sin^{-1}(\cos(2)) \), we will simplify each term step by step.

### Step 1: Simplifying \( \sin^{-1}(\sin(22\pi/7)) \)

1. **Find the equivalent angle**:
\[ 22\pi/7 = 3\pi + \pi/7 \]
Since \( \sin(x) \) is periodic with period \( 2\pi \), we can reduce \( 22\pi/7 \) to:
\[ 22\pi/7 - 2\pi = 22\pi/7 - 14\pi/7 = 8\pi/7 \]
Now, \( \sin(22\pi/7) = \sin(8\pi/7) \).

2. **Determine the value of \( \sin^{-1}(\sin(8\pi/7)) \)**:
Since \( 8\pi/7 \) is in the third quadrant, we have:
\[ \sin^{-1}(\sin(8\pi/7)) = \pi - 8\pi/7 = -\pi/7 \]

### Step 2: Simplifying \( \cos^{-1}(\cos(5\pi/3)) \)

1. **Find the equivalent angle**:
\[ 5\pi/3 = 2\pi - \pi/3 \]
Thus, \( \cos(5\pi/3) = \cos(-\pi/3) = \cos(\pi/3) \).

2. **Determine the value of \( \cos^{-1}(\cos(5\pi/3)) \)**:
Since \( \cos^{-1} \) gives the principal value in the range \( [0, \pi] \):
\[ \cos^{-1}(\cos(5\pi/3)) = \pi/3 \]

### Step 3: Simplifying \( \tan^{-1}(\tan(5\pi/7)) \)

1. **Determine the value of \( \tan^{-1}(\tan(5\pi/7)) \)**:
Since \( 5\pi/7 \) is in the second quadrant:
\[ \tan^{-1}(\tan(5\pi/7)) = \pi - 5\pi/7 = 2\pi/7 \]

### Step 4: Simplifying \( \sin^{-1}(\cos(2)) \)

1. **Use the identity**:
\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \]
Thus:
\[ \sin^{-1}(\cos(2)) = \frac{\pi}{2} - 2 \]

### Step 5: Combine all parts

Now, we combine all the simplified parts:
\[ \sin^{-1}(\sin(22\pi/7)) + \cos^{-1}(\cos(5\pi/3)) + \tan^{-1}(\tan(5\pi/7)) + \sin^{-1}(\cos(2)) \]
Substituting the values we found:
\[ -\frac{\pi}{7} + \frac{\pi}{3} + \frac{2\pi}{7} + \left(\frac{\pi}{2} - 2\right) \]

### Step 6: Finding a common denominator

The common denominator for \( 7, 3, 2 \) is \( 42 \):
1. Convert each term:
\[ -\frac{\pi}{7} = -\frac{6\pi}{42}, \quad \frac{\pi}{3} = \frac{14\pi}{42}, \quad \frac{2\pi}{7} = \frac{12\pi}{42}, \quad \frac{\pi}{2} = \frac{21\pi}{42} \]
2. Combine:
\[ -\frac{6\pi}{42} + \frac{14\pi}{42} + \frac{12\pi}{42} + \frac{21\pi}{42} - 2 \]
\[ = \frac{41\pi}{42} - 2 \]

### Final Result

Thus, the final value of the expression is:
\[ \frac{41\pi}{42} - 2 \]