`sin^(-1){sqrt((1-cosx)/(2))}`

Answer

To solve the problem \( y = \sin^{-1}\left(\sqrt{\frac{1 - \cos x}{2}}\right) \), we will differentiate \( y \) with respect to \( x \). Let's go through the steps systematically.

### Step 1: Rewrite the expression inside the inverse sine function

We know from trigonometric identities that:
\[ 1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right) \]
Thus, we can rewrite the expression as:
\[ \sqrt{\frac{1 - \cos x}{2}} = \sqrt{\frac{2 \sin^2\left(\frac{x}{2}\right)}{2}} = \sqrt{\sin^2\left(\frac{x}{2}\right)} = \sin\left(\frac{x}{2}\right) \]
So, we can rewrite \( y \):
\[ y = \sin^{-1}\left(\sin\left(\frac{x}{2}\right)\right) \]

### Step 2: Simplify the expression for \( y \)

Since \( \sin^{-1}(\sin(\theta)) = \theta \) for \( \theta \) in the range of \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), we have:
\[ y = \frac{x}{2} \]

### Step 3: Differentiate \( y \) with respect to \( x \)

Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{2} \]

### Final Result

Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{2} \]

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