The number of integers for which the equation `sin^(-1)x+cos^(-1)x+tan^(-1)x=n` has real solution(s) is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( \sin^{-1} x + \cos^{-1} x + \tan^{-1} x = n \) for the number of integers \( n \) that yield real solutions, we can follow these steps:

### Step 1: Simplify the Equation
We know from trigonometric identities that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
Thus, we can rewrite the equation as:
\[ \frac{\pi}{2} + \tan^{-1} x = n \]

### Step 2: Isolate \( \tan^{-1} x \)
Rearranging the equation gives us:
\[ \tan^{-1} x = n - \frac{\pi}{2} \]

### Step 3: Determine the Range of \( \tan^{-1} x \)
The function \( \tan^{-1} x \) has a range of \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \). Therefore, we need to find the values of \( n \) such that:
\[ -\frac{\pi}{2} < n - \frac{\pi}{2} < \frac{\pi}{2} \]

### Step 4: Solve the Inequalities
1. From the left inequality:
\[ -\frac{\pi}{2} < n - \frac{\pi}{2} \implies n > 0 \]

2. From the right inequality:
\[ n - \frac{\pi}{2} < \frac{\pi}{2} \implies n < \pi \]

### Step 5: Determine the Integer Values of \( n \)
Now we have the range for \( n \):
\[ 0 < n < \pi \]
Given that \( \pi \approx 3.14 \), the integers satisfying this inequality are:
- \( n = 1 \)
- \( n = 2 \)
- \( n = 3 \)

### Step 6: Count the Integer Solutions
The integers \( n \) that satisfy the equation are \( 1, 2, 3 \). Thus, there are **3 integers** in total.

### Final Answer
The number of integers for which the equation \( \sin^{-1} x + \cos^{-1} x + \tan^{-1} x = n \) has real solutions is **3**.

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