The number of integral values of `k` for which the equation `sin^(-1)x+tan^(-1)x=2k+1` has a solution is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to analyze the equation:

\[ \sin^{-1}x + \tan^{-1}x = 2k + 1 \]

### Step 1: Determine the Range of \(\sin^{-1}x\) and \(\tan^{-1}x\)

The function \(\sin^{-1}x\) is defined for \(x \in [-1, 1]\) and its range is:

\[ \sin^{-1}x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]

The function \(\tan^{-1}x\) is defined for all real \(x\) and its range is:

\[ \tan^{-1}x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]

### Step 2: Find the Range of the Sum \(\sin^{-1}x + \tan^{-1}x\)

The sum of these two functions will have a range determined by their individual ranges. The minimum value occurs when both functions are at their minimum:

\[ -\frac{\pi}{2} + \left(-\frac{\pi}{2}\right) = -\pi \]

The maximum value occurs when both functions are at their maximum:

\[ \frac{\pi}{2} + \frac{\pi}{2} = \pi \]

Thus, the range of \(\sin^{-1}x + \tan^{-1}x\) is:

\[ \left[-\pi, \pi\right] \]

### Step 3: Set the Equation and Analyze

We need to find out for which integer values of \(k\) the equation:

\[ \sin^{-1}x + \tan^{-1}x = 2k + 1 \]

has a solution. This means \(2k + 1\) must lie within the range \([- \pi, \pi]\).

### Step 4: Solve for \(k\)

To find the bounds for \(k\), we set up the inequalities:

\[ -\pi \leq 2k + 1 \leq \pi \]

Subtracting 1 from all parts:

\[ -\pi - 1 \leq 2k \leq \pi - 1 \]

Dividing the entire inequality by 2:

\[ \frac{-\pi - 1}{2} \leq k \leq \frac{\pi - 1}{2} \]

### Step 5: Calculate the Integral Values of \(k\)

Now we need to calculate the approximate values of \(\frac{-\pi - 1}{2}\) and \(\frac{\pi - 1}{2}\):

Using \(\pi \approx 3.14\):

\[ \frac{-3.14 - 1}{2} \approx \frac{-4.14}{2} \approx -2.07 \]

\[ \frac{3.14 - 1}{2} \approx \frac{2.14}{2} \approx 1.07 \]

Thus, the integer values of \(k\) that satisfy this inequality are:

\[ k = -2, -1, 0, 1 \]

### Conclusion

The total number of integral values of \(k\) is:

\[ \text{Total values} = 4 \quad (k = -2, -1, 0, 1) \]

### Final Answer

The number of integral values of \(k\) for which the equation has a solution is **4**.

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