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 means that \( \sin \theta = x \).

### Step 2: Create a right triangle
In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side (perpendicular) to the length of the hypotenuse. Therefore, we can represent this as:
- Opposite side (perpendicular) = \( x \)
- Hypotenuse = \( 1 \)

### Step 3: Find the length of the adjacent side
Using the Pythagorean theorem, we can find the length of the adjacent side (base):
\[ \text{Adjacent}^2 + \text{Opposite}^2 = \text{Hypotenuse}^2 \]
Substituting the known values:
\[ \text{Adjacent}^2 + x^2 = 1^2 \]
\[ \text{Adjacent}^2 + x^2 = 1 \]
\[ \text{Adjacent}^2 = 1 - x^2 \]
\[ \text{Adjacent} = \sqrt{1 - x^2} \]

### Step 4: Calculate \( \cot \theta \)
The cotangent of an angle is defined as the ratio of the length of the adjacent side to the length of the opposite side:
\[ \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{\sqrt{1 - x^2}}{x} \]

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

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

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