The principal value of
`cos^(-1)((1)/(2))+2sin^(-1)((1)/(2))+4tan^(-1)((1)/(sqrt3))` is

Options

A. `(pi)/(3)`
B. `(pi)/(6)`
C. `(4pi)/(3)`
D. `(3pi)/(4)`

Answer

To solve the expression \( \cos^{-1}\left(\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right) + 4\tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \cos^{-1}\left(\frac{1}{2}\right) \)
The value of \( \cos^{-1}\left(\frac{1}{2}\right) \) is known to be \( \frac{\pi}{3} \) radians.

### Step 2: Evaluate \( \sin^{-1}\left(\frac{1}{2}\right) \)
The value of \( \sin^{-1}\left(\frac{1}{2}\right) \) is known to be \( \frac{\pi}{6} \) radians. Since we have \( 2\sin^{-1}\left(\frac{1}{2}\right) \), we multiply this by 2:
\[ 2\sin^{-1}\left(\frac{1}{2}\right) = 2 \times \frac{\pi}{6} = \frac{\pi}{3} \]

### Step 3: Evaluate \( \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \)
The value of \( \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \) is known to be \( \frac{\pi}{6} \) radians. Since we have \( 4\tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \), we multiply this by 4:
\[ 4\tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = 4 \times \frac{\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3} \]

### Step 4: Combine all the values
Now we combine all the evaluated parts:
\[ \cos^{-1}\left(\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right) + 4\tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{3} + \frac{\pi}{3} + \frac{2\pi}{3} \]

### Step 5: Simplify the expression
Adding these fractions:
\[ \frac{\pi}{3} + \frac{\pi}{3} + \frac{2\pi}{3} = \frac{1\pi + 1\pi + 2\pi}{3} = \frac{4\pi}{3} \]

### Final Answer
Thus, the principal value of the given expression is:
\[ \boxed{\frac{4\pi}{3}} \]