If ` f ( x) = cos^(-1) ( cos ( x + 1) ) " and " g(x) = sin ^(-1) ( sin (x + 2))`, then

Options

A. `f(1) + g (1) = ( pi -1)`
B. `f (1) gt g(1)`
C. ` f(2) gt g (2) `
D. `f(2) lt g (2)`

Answer

To solve the problem, we need to evaluate the functions \( f(x) \) and \( g(x) \) at specific points and analyze the results.

### Step 1: Evaluate \( f(1) \)
Given:
\[ f(x) = \cos^{-1}(\cos(x + 1)) \]
We need to find \( f(1) \):
\[ f(1) = \cos^{-1}(\cos(1 + 1)) = \cos^{-1}(\cos(2)) \]
Since \( 2 \) is in the range of \( \cos^{-1} \) (which is \( [0, \pi] \)), we have:
\[ f(1) = 2 \]

### Step 2: Evaluate \( f(2) \)
Now, we find \( f(2) \):
\[ f(2) = \cos^{-1}(\cos(2 + 1)) = \cos^{-1}(\cos(3)) \]
Again, since \( 3 \) is in the range of \( \cos^{-1} \), we have:
\[ f(2) = 3 \]

### Step 3: Evaluate \( g(1) \)
Next, we evaluate \( g(1) \):
\[ g(x) = \sin^{-1}(\sin(x + 2)) \]
Thus,
\[ g(1) = \sin^{-1}(\sin(1 + 2)) = \sin^{-1}(\sin(3)) \]
Since \( 3 \) is greater than \( \frac{\pi}{2} \) but less than \( \pi \), we have:
\[ g(1) = \pi - 3 \]

### Step 4: Evaluate \( g(2) \)
Now, we find \( g(2) \):
\[ g(2) = \sin^{-1}(\sin(2 + 2)) = \sin^{-1}(\sin(4)) \]
Since \( 4 \) is greater than \( \pi \), we can write:
\[ g(2) = \pi - 4 \]

### Step 5: Compare values
Now we can compare the values we have calculated:
- \( f(1) = 2 \)
- \( g(1) = \pi - 3 \)
- \( f(2) = 3 \)
- \( g(2) = \pi - 4 \)

### Step 6: Analyze the options
1. **Option A**: \( f(1) + g(1) = 2 + (\pi - 3) = \pi - 1 \) (Correct)
2. **Option B**: \( f(1) > g(1) \) means \( 2 > \pi - 3 \). Since \( \pi \approx 3.14 \), \( 2 > -0.14 \) (Correct)
3. **Option C**: \( f(2) > g(2) \) means \( 3 > \pi - 4 \). Since \( 3 > -0.14 \) (Correct)
4. **Option D**: \( f(2) < g(2) \) means \( 3 < \pi - 4 \). Since \( 3 < -0.14 \) is incorrect (Incorrect)

### Conclusion
The correct options are A, B, and C. Option D is incorrect.