The value of `cos^(-1){sin(cos^(-1)(1/2))}` is

Options

A. `(pi)/(3)`
B. `-(pi)/(3)`
C. `(pi)/(6)`
D. `(5pi)/(6)`

Explanation

To solve the problem \( \cos^{-1}(\sin(\cos^{-1}(1/2))) \), we will follow these steps:

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

The value of \( \cos^{-1}(1/2) \) corresponds to the angle whose cosine is \( 1/2 \). We know that:

\[ \cos(\pi/3) = 1/2 \]

Thus,

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

### Step 2: Substitute into the sine function

Now we substitute this value into the sine function:

\[ \sin(\cos^{-1}(1/2)) = \sin\left(\frac{\pi}{3}\right) \]

### Step 3: Evaluate \( \sin(\frac{\pi}{3}) \)

The value of \( \sin(\frac{\pi}{3}) \) is:

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

### Step 4: Substitute into the inverse cosine function

Now, we need to find:

\[ \cos^{-1}\left(\sin\left(\cos^{-1}(1/2)\right)\right) = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) \]

### Step 5: Evaluate \( \cos^{-1}(\frac{\sqrt{3}}{2}) \)

The value of \( \cos^{-1}(\frac{\sqrt{3}}{2}) \) corresponds to the angle whose cosine is \( \frac{\sqrt{3}}{2} \). We know that:

\[ \cos(\pi/6) = \frac{\sqrt{3}}{2} \]

Thus,

\[ \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6} \]

### Final Answer

Therefore, the value of \( \cos^{-1}(\sin(\cos^{-1}(1/2))) \) is:

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

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