Solution(s) of the equation
`sin(-pi/3+tan^(-1)x+cot^(-1)x)=1/2` is/are

Options

A. `1/2`
B. `-1/2`
C. `sqrt3`
D. `-sqrt3`

Answer

To solve the equation \( \sin\left(-\frac{\pi}{3} + \tan^{-1}x + \cot^{-1}x\right) = \frac{1}{2} \), we will follow these steps:

### Step 1: Simplify the expression involving inverse trigonometric functions
We know that:
\[ \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \]
for all \( x \) in the domain of real numbers. Therefore, we can rewrite the equation as:
\[ \sin\left(-\frac{\pi}{3} + \frac{\pi}{2}\right) = \frac{1}{2} \]

### Step 2: Simplify the angle
Now, simplify the angle:
\[ -\frac{\pi}{3} + \frac{\pi}{2} = -\frac{2\pi}{6} + \frac{3\pi}{6} = \frac{\pi}{6} \]
So, the equation becomes:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

### Step 3: Evaluate the sine function
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Thus, the equation holds true.

### Step 4: Determine the values of \( x \)
Since the equation is satisfied for all \( x \) in the domain of real numbers, we conclude that:
\[ \text{All values of } x \text{ are solutions.} \]

### Step 5: Conclusion
Therefore, the solution(s) of the equation is:
\[ \text{All } x \in (-\infty, \infty) \]

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