The value of the expression `sin[cot^(-1)1)}]` is

Options

A. `1/sqrt2`
B. `sqrt(3/2)`
C. 1
D. 0

Answer

To find the value of the expression \( \sin(\cot^{-1}(1)) \), we can follow these steps:

### Step 1: Understand the Inverse Cotangent
Let \( \theta = \cot^{-1}(1) \). This means that \( \cot(\theta) = 1 \).

**Hint**: Recall that \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \).

### Step 2: Determine the Angle
The cotangent of an angle is 1 when the angle is \( \frac{\pi}{4} \) radians (or 45 degrees). Therefore, we have:
\[ \theta = \frac{\pi}{4} \]

**Hint**: Remember that \( \cot(\frac{\pi}{4}) = 1 \).

### Step 3: Find the Sine of the Angle
Now, we need to find \( \sin(\theta) \):
\[ \sin(\theta) = \sin\left(\frac{\pi}{4}\right) \]

**Hint**: Recall the sine value for common angles.

### Step 4: Calculate the Sine Value
The sine of \( \frac{\pi}{4} \) is:
\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

**Hint**: Use the unit circle or the properties of a 45-45-90 triangle to find this value.

### Step 5: Conclusion
Thus, the value of the expression \( \sin(\cot^{-1}(1)) \) is:
\[ \sin(\cot^{-1}(1)) = \frac{1}{\sqrt{2}} \]

**Final Answer**: \( \frac{1}{\sqrt{2}} \)