The value of `2sec^(-1) 2 + 2 sin^(-1)(1/2)` is

Options

A. `(pi)/(6)`
B. `(5pi)/(6)`
C. `(7pi)/(6)`
D. `1`

Explanation

To solve the expression \(2 \sec^{-1}(2) + 2 \sin^{-1}\left(\frac{1}{2}\right)\), we will break it down step by step.

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

The secant function is the reciprocal of the cosine function. Therefore, if \( y = \sec^{-1}(2) \), then:

\[ \sec(y) = 2 \implies \cos(y) = \frac{1}{2} \]

The angle \( y \) for which \( \cos(y) = \frac{1}{2} \) is:

\[ y = \frac{\pi}{3} \]

Thus,

\[ \sec^{-1}(2) = \frac{\pi}{3} \]

### Step 2: Evaluate \( \sin^{-1}\left(\frac{1}{2}\right) \)

The sine function gives \( \frac{1}{2} \) at:

\[ x = \frac{\pi}{6} \]

So,

\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

### Step 3: Substitute the values back into the expression

Now we substitute these values back into the original expression:

\[ 2 \sec^{-1}(2) + 2 \sin^{-1}\left(\frac{1}{2}\right) = 2 \left(\frac{\pi}{3}\right) + 2 \left(\frac{\pi}{6}\right) \]

### Step 4: Simplify the expression

Calculating each term:

\[ 2 \left(\frac{\pi}{3}\right) = \frac{2\pi}{3} \]
\[ 2 \left(\frac{\pi}{6}\right) = \frac{\pi}{3} \]

Now, adding these two results together:

\[ \frac{2\pi}{3} + \frac{\pi}{3} = \frac{2\pi + \pi}{3} = \frac{3\pi}{3} = \pi \]

### Final Answer

Thus, the value of \( 2 \sec^{-1}(2) + 2 \sin^{-1}\left(\frac{1}{2}\right) \) is:

\[ \boxed{\pi} \]