The value of `cos^(-1) (cos 12) - sin^(-1)(sin 14)` is

Options

A. 2
B. `8pi - 26`
C. `4pi + 2`
D. None of these

Explanation

To solve the expression \( \cos^{-1}(\cos 12) - \sin^{-1}(\sin 14) \), we will follow these steps:

### Step 1: Evaluate \( \cos^{-1}(\cos 12) \)

The function \( \cos^{-1}(x) \) returns the angle whose cosine is \( x \). The range of \( \cos^{-1}(x) \) is \( [0, \pi] \). Since \( 12 \) is greater than \( \pi \) (approximately \( 3.14 \)), we need to find an equivalent angle within the range.

Using the property:
\[ \cos^{-1}(\cos \theta) = 2\pi - \theta \quad \text{if } \theta > \pi \]
we can rewrite:
\[ \cos^{-1}(\cos 12) = 2\pi - 12 \]

### Step 2: Evaluate \( \sin^{-1}(\sin 14) \)

The function \( \sin^{-1}(x) \) returns the angle whose sine is \( x \). The range of \( \sin^{-1}(x) \) is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). Since \( 14 \) is greater than \( \frac{\pi}{2} \), we need to find an equivalent angle within the range.

Using the property:
\[ \sin^{-1}(\sin \theta) = \pi - \theta \quad \text{if } \theta > \frac{\pi}{2} \text{ and } \theta < \frac{3\pi}{2} \]
we can rewrite:
\[ \sin^{-1}(\sin 14) = \pi - 14 \]

### Step 3: Substitute back into the expression

Now we substitute back into the original expression:
\[ \cos^{-1}(\cos 12) - \sin^{-1}(\sin 14) = (2\pi - 12) - (\pi - 14) \]

### Step 4: Simplify the expression

Now we simplify:
\[ = 2\pi - 12 - \pi + 14 \]
\[ = (2\pi - \pi) + (14 - 12) \]
\[ = \pi + 2 \]

### Step 5: Final expression

Thus, the final value of the expression is:
\[ \pi + 2 \]