`cos^(-1)(-x),absx le 1`, is equal to

Options

A. `-cos^(-1)x`
B. `cos^(-1)x`
C. `pi/2-cos^(-1)x`
D. `pi/2+sin^(-1)x`

Answer

To solve the expression \( \cos^{-1}(-x) \) where \( |x| \leq 1 \), we can follow these steps:

### Step 1: Understand the properties of the inverse cosine function
The range of the function \( \cos^{-1}(x) \) is from \( 0 \) to \( \pi \), and its domain is \( [-1, 1] \). Since \( |x| \leq 1 \), \( -x \) will also lie within the domain of the inverse cosine function.

**Hint:** Recall the range and domain of the inverse cosine function.

### Step 2: Use the property of inverse cosine
We can use the property of the inverse cosine function that states:
\[ \cos^{-1}(-x) = \pi - \cos^{-1}(x) \]
This property helps us relate the cosine of a negative angle to the cosine of a positive angle.

**Hint:** Look for properties of inverse trigonometric functions that relate angles.

### Step 3: Substitute the property into the equation
Using the property from Step 2, we can substitute:
\[ \cos^{-1}(-x) = \pi - \cos^{-1}(x) \]

**Hint:** Substitute the known property directly into your equation.

### Step 4: Use the relationship between sine and cosine
We also know that:
\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \]
From this, we can express \( \cos^{-1}(x) \) as:
\[ \cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(x) \]

**Hint:** Remember the relationship between sine and cosine inverse functions.

### Step 5: Substitute \( \cos^{-1}(x) \) into the equation
Now, substituting \( \cos^{-1}(x) \) into our equation from Step 3:
\[ \cos^{-1}(-x) = \pi - \left(\frac{\pi}{2} - \sin^{-1}(x)\right) \]
This simplifies to:
\[ \cos^{-1}(-x) = \pi - \frac{\pi}{2} + \sin^{-1}(x) = \frac{\pi}{2} + \sin^{-1}(x) \]

**Hint:** Simplify the expression carefully to find the final result.

### Final Result
Thus, we conclude that:
\[ \cos^{-1}(-x) = \frac{\pi}{2} + \sin^{-1}(x) \]

### Summary
The final expression for \( \cos^{-1}(-x) \) where \( |x| \leq 1 \) is:
\[ \cos^{-1}(-x) = \frac{\pi}{2} + \sin^{-1}(x) \]