Find the value of `sin^(-1)(cos(sin^(-1)x))+cos^(-1)(sin(cos^(-1)x))`

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `(3pi)/(4)`
D. 0

Explanation

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

### Step 1: Rewrite the Inverse Functions
We start by rewriting the inverse trigonometric functions in terms of each other. We know that:
\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \]
Thus, we can express \( \sin^{-1}(x) \) as:
\[ \sin^{-1}(x) = \frac{\pi}{2} - \cos^{-1}(x) \]

### Step 2: Substitute in the Expression
Now, we can substitute \( \sin^{-1}(x) \) in our expression:
\[ \sin^{-1}(\cos(\sin^{-1}x)) + \cos^{-1}(\sin(\cos^{-1}x)) \]
We can rewrite \( \sin^{-1}(x) \) in terms of \( \cos^{-1}(x) \):
\[ = \sin^{-1}(\cos(\frac{\pi}{2} - \cos^{-1}(x))) + \cos^{-1}(\sin(\cos^{-1}(x))) \]

### Step 3: Simplify the Cosine and Sine Functions
Using the identity \( \cos(\frac{\pi}{2} - \theta) = \sin(\theta) \):
\[ = \sin^{-1}(\sin(\cos^{-1}(x))) + \cos^{-1}(\sin(\cos^{-1}(x))) \]

### Step 4: Use the Properties of Inverse Functions
Now, we know that:
\[ \sin^{-1}(\sin(\theta)) = \theta \quad \text{for } \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
and
\[ \cos^{-1}(\sin(\theta)) = \frac{\pi}{2} - \theta \quad \text{for } \theta \in [0, \frac{\pi}{2}] \]
Thus, we can simplify:
\[ = \cos^{-1}(x) + \left(\frac{\pi}{2} - \cos^{-1}(x)\right) \]

### Step 5: Combine the Terms
Now, combining the terms gives us:
\[ = \frac{\pi}{2} \]

### Final Result
Therefore, the value of the expression \( \sin^{-1}(\cos(\sin^{-1}x)) + \cos^{-1}(\sin(\cos^{-1}x)) \) is:
\[ \frac{\pi}{2} \]

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