The value of `cot(sin^(-1)x)` is :

Options

A. `(sqrt(1+x^(2)))/(x)`
B. `(x)/(sqrt(1+x^(2)))`
C. `(1)/(x)`
D. `(sqrt(1-x^(2)))/(x)`

Answer

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

### Step 1: Define the angle
Let \( \theta = \sin^{-1} x \). This implies that:
\[ \sin \theta = x \]

### Step 2: Construct a right triangle
We can represent this relationship using a right triangle. In this triangle:
- The angle \( \theta \) is opposite the side of length \( x \) (the opposite side).
- The hypotenuse has a length of \( 1 \) (since \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)).

### Step 3: Find the length of the adjacent side
Using the Pythagorean theorem, we can find the length of the adjacent side (let's denote it as \( b \)):
\[ b^2 + x^2 = 1^2 \]
\[ b^2 = 1 - x^2 \]
\[ b = \sqrt{1 - x^2} \]

### Step 4: Calculate \( \cot \theta \)
The cotangent of \( \theta \) is given by:
\[ \cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{b}{x} = \frac{\sqrt{1 - x^2}}{x} \]

### Step 5: Conclusion
Thus, we find that:
\[ \cot(\sin^{-1} x) = \frac{\sqrt{1 - x^2}}{x} \]

### Final Answer
The value of \( \cot(\sin^{-1} x) \) is \( \frac{\sqrt{1 - x^2}}{x} \).

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