The range of values of p for which the equation `sin cos^(-1) ( cos( tan^(-1) x)) = p` has a solution is

Options

A. `( - 1/sqrt2, 1/sqrt2]`
B. `[0, 1)`
C. `[ 1/sqrt2 , 1)`
D. `(-1,1)`

Answer

To solve the equation \( \sin(\cos^{-1}(\cos(\tan^{-1} x))) = p \) and find the range of values of \( p \) for which this equation has a solution, we can follow these steps:

### Step 1: Understand the Components
We start with the expression \( \tan^{-1}(x) \). The range of \( \tan^{-1}(x) \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

### Step 2: Evaluate \( \cos(\tan^{-1}(x)) \)
Using the identity for cosine in terms of tangent, we have:
\[ \cos(\tan^{-1}(x)) = \frac{1}{\sqrt{1 + x^2}} \]
This is because if \( \theta = \tan^{-1}(x) \), then \( \tan(\theta) = x \) implies \( \sin(\theta) = \frac{x}{\sqrt{1+x^2}} \) and \( \cos(\theta) = \frac{1}{\sqrt{1+x^2}} \).

### Step 3: Evaluate \( \cos^{-1}(\cos(\tan^{-1}(x))) \)
Now we need to find \( \cos^{-1}(\cos(\tan^{-1}(x))) \). Since \( \tan^{-1}(x) \) is in the range \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), we have:
\[ \cos^{-1}(\cos(\tan^{-1}(x))) = \tan^{-1}(x) \]
This holds because \( \tan^{-1}(x) \) is within the principal range of the \( \cos^{-1} \) function.

### Step 4: Evaluate \( \sin(\cos^{-1}(\cos(\tan^{-1}(x)))) \)
Now we can substitute back into our original equation:
\[ \sin(\cos^{-1}(\cos(\tan^{-1}(x)))) = \sin(\tan^{-1}(x)) \]
Using the identity for sine in terms of tangent:
\[ \sin(\tan^{-1}(x)) = \frac{x}{\sqrt{1+x^2}} \]

### Step 5: Set the Equation
Now we have:
\[ \frac{x}{\sqrt{1+x^2}} = p \]

### Step 6: Find the Range of \( p \)
To find the range of \( p \), we analyze the function \( f(x) = \frac{x}{\sqrt{1+x^2}} \).

1. As \( x \to -\infty \), \( f(x) \to -1 \).
2. As \( x \to \infty \), \( f(x) \to 1 \).
3. At \( x = 0 \), \( f(0) = 0 \).

The function \( f(x) \) is continuous and increases from -1 to 1 as \( x \) varies from \(-\infty\) to \(\infty\).

### Conclusion
Thus, the range of values for \( p \) is:
\[ p \in (-1, 1) \]
Since the endpoints are not included, we can express this as:
\[ p \in (-1, 1) \]

### Final Answer
The range of values of \( p \) for which the equation has a solution is:
\[ (-1, 1) \]