Solve the following equations :
`cos (sin^(-1) x) = 1/2 `

Answer

To solve the equation \( \cos(\sin^{-1} x) = \frac{1}{2} \), we can follow these steps:

### Step 1: Understand the Inverse Function
The term \( \sin^{-1} x \) represents the angle \( \theta \) such that \( \sin \theta = x \). The range of \( \theta \) for \( \sin^{-1} x \) is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

### Step 2: Rewrite the Equation
Using the identity for cosine in terms of sine, we can rewrite:
\[ \cos(\sin^{-1} x) = \sqrt{1 - \sin^2(\sin^{-1} x)} = \sqrt{1 - x^2} \]
Thus, the equation becomes:
\[ \sqrt{1 - x^2} = \frac{1}{2} \]

### Step 3: Square Both Sides
To eliminate the square root, we square both sides:
\[ 1 - x^2 = \left(\frac{1}{2}\right)^2 \]
\[ 1 - x^2 = \frac{1}{4} \]

### Step 4: Solve for \( x^2 \)
Rearranging the equation gives:
\[ x^2 = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 5: Take the Square Root
Taking the square root of both sides, we find:
\[ x = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \]

### Step 6: Write the Final Answer
Thus, the solutions to the equation \( \cos(\sin^{-1} x) = \frac{1}{2} \) are:
\[ x = \frac{\sqrt{3}}{2} \quad \text{and} \quad x = -\frac{\sqrt{3}}{2} \]