If `x = tan^(-1) 1 - cos^(-1) ( - 1/2) + sin^(-1) 1/2 , y = cos (1/2 cos^(-1) (1/8))`, then

Options

A. ` x = pi y`
B. ` y = pi x`
C. ` tan x = - (4//3) y`
D. ` tan x = ( 4//3) y`

Answer

To solve the problem, we need to evaluate the expressions for \( x \) and \( y \) given in the question and then check which of the provided options is correct.

### Step 1: Evaluate \( x \)

Given:
\[ x = \tan^{-1}(1) - \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(\frac{1}{2}\right) \]

1. **Calculate \( \tan^{-1}(1) \)**:
\[ \tan^{-1}(1) = \frac{\pi}{4} \]

2. **Calculate \( \cos^{-1}\left(-\frac{1}{2}\right) \)**:
\[ \cos^{-1}\left(-\frac{1}{2}\right) = \pi - \cos^{-1}\left(\frac{1}{2}\right) = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

3. **Calculate \( \sin^{-1}\left(\frac{1}{2}\right) \)**:
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

4. **Substituting values into \( x \)**:
\[ x = \frac{\pi}{4} - \frac{2\pi}{3} + \frac{\pi}{6} \]

5. **Finding a common denominator (12)**:
\[ x = \frac{3\pi}{12} - \frac{8\pi}{12} + \frac{2\pi}{12} = \frac{3\pi - 8\pi + 2\pi}{12} = \frac{-3\pi}{12} = -\frac{\pi}{4} \]

### Step 2: Evaluate \( y \)

Given:
\[ y = \cos\left(\frac{1}{2} \cos^{-1}\left(\frac{1}{8}\right)\right) \]

1. **Let \( \theta = \cos^{-1}\left(\frac{1}{8}\right) \)**:
\[ \cos(\theta) = \frac{1}{8} \]

2. **Using the half-angle formula**:
\[ y = \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} = \sqrt{\frac{1 + \frac{1}{8}}{2}} = \sqrt{\frac{\frac{9}{8}}{2}} = \sqrt{\frac{9}{16}} = \frac{3}{4} \]

### Step 3: Check the options

Now we have:
- \( x = -\frac{\pi}{4} \)
- \( y = \frac{3}{4} \)

1. **Option A: \( x = \pi y \)**:
\[ -\frac{\pi}{4} \neq \pi \cdot \frac{3}{4} \quad \text{(not true)} \]

2. **Option B: \( y = \pi x \)**:
\[ \frac{3}{4} \neq \pi \cdot \left(-\frac{\pi}{4}\right) \quad \text{(not true)} \]

3. **Option C: \( \tan(x) = -\frac{4}{3}y \)**:
\[ \tan\left(-\frac{\pi}{4}\right) = -1 \quad \text{and} \quad -\frac{4}{3} \cdot \frac{3}{4} = -1 \quad \text{(true)} \]

4. **Option D: \( \tan(x) = \frac{4}{3}y \)**:
\[ -1 \neq \frac{4}{3} \cdot \frac{3}{4} = 1 \quad \text{(not true)} \]

### Conclusion

The correct option is:
\[ \text{Option C: } \tan(x) = -\frac{4}{3}y \]