The value of `sin{tan^(-1) (tan ((7pi)/6) + cos^(-1)(cos ((7pi)/3))}` is _____.

Options

A. 0
B. 1
C. -1
D. none of these

Answer

To solve the problem, we need to evaluate the expression:

\[ \sin\left(\tan^{-1}\left(\tan\left(\frac{7\pi}{6}\right)\right) + \cos^{-1}\left(\cos\left(\frac{7\pi}{3}\right)\right)\right) \]

Let's break it down step by step.

### Step 1: Simplify \( \tan^{-1}\left(\tan\left(\frac{7\pi}{6}\right)\right) \)

1. First, we note that \( \frac{7\pi}{6} \) can be rewritten as:
\[ \frac{7\pi}{6} = \pi + \frac{\pi}{6} \]

2. The tangent function has a periodicity of \( \pi \), so:
\[ \tan\left(\frac{7\pi}{6}\right) = \tan\left(\pi + \frac{\pi}{6}\right) = \tan\left(\frac{\pi}{6}\right) \]

3. Now, we apply the inverse tangent function:
\[ \tan^{-1}\left(\tan\left(\frac{7\pi}{6}\right)\right) = \tan^{-1}\left(\tan\left(\frac{\pi}{6}\right)\right) \]

4. Since \( \frac{\pi}{6} \) is in the range of \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), we have:
\[ \tan^{-1}\left(\tan\left(\frac{\pi}{6}\right)\right) = \frac{\pi}{6} \]

### Step 2: Simplify \( \cos^{-1}\left(\cos\left(\frac{7\pi}{3}\right)\right) \)

1. Next, we simplify \( \frac{7\pi}{3} \):
\[ \frac{7\pi}{3} = 2\pi + \frac{\pi}{3} \]

2. The cosine function also has a periodicity of \( 2\pi \), so:
\[ \cos\left(\frac{7\pi}{3}\right) = \cos\left(2\pi + \frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) \]

3. Now, we apply the inverse cosine function:
\[ \cos^{-1}\left(\cos\left(\frac{7\pi}{3}\right)\right) = \cos^{-1}\left(\cos\left(\frac{\pi}{3}\right)\right) \]

4. Since \( \frac{\pi}{3} \) is in the range of \( [0, \pi] \), we have:
\[ \cos^{-1}\left(\cos\left(\frac{\pi}{3}\right)\right) = \frac{\pi}{3} \]

### Step 3: Combine the results

Now we can combine the results from Step 1 and Step 2:
\[ \sin\left(\tan^{-1}\left(\tan\left(\frac{7\pi}{6}\right)\right) + \cos^{-1}\left(\cos\left(\frac{7\pi}{3}\right)\right)\right) = \sin\left(\frac{\pi}{6} + \frac{\pi}{3}\right) \]

### Step 4: Calculate \( \frac{\pi}{6} + \frac{\pi}{3} \)

1. Convert \( \frac{\pi}{3} \) to a common denominator:
\[ \frac{\pi}{3} = \frac{2\pi}{6} \]

2. Now add:
\[ \frac{\pi}{6} + \frac{2\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \]

### Step 5: Calculate \( \sin\left(\frac{\pi}{2}\right) \)

Finally, we calculate:
\[ \sin\left(\frac{\pi}{2}\right) = 1 \]

### Final Answer

Thus, the value of the expression is:
\[ \boxed{1} \]