`SinA=1//x` then `CosA=` ?

Options

A. `x2+1sqrt'''x X2+1x`
B. `"xx"2-1sqrt'''x X2-1`
C. `x2-1sqrt'''x X2-1x`
D. `"xx"2+1sqrt''' "xx"2+1`

Answer

To find the value of \( \cos A \) given that \( \sin A = \frac{1}{x} \), we can use the Pythagorean identity which states that:

\[ \sin^2 A + \cos^2 A = 1 \]

### Step-by-step Solution:

1. **Start with the Pythagorean Identity**:
\[ \sin^2 A + \cos^2 A = 1 \]

2. **Substitute the value of \( \sin A \)**:
Since \( \sin A = \frac{1}{x} \), we can substitute this into the identity:
\[ \left(\frac{1}{x}\right)^2 + \cos^2 A = 1 \]

3. **Calculate \( \sin^2 A \)**:
\[ \frac{1}{x^2} + \cos^2 A = 1 \]

4. **Isolate \( \cos^2 A \)**:
To find \( \cos^2 A \), we rearrange the equation:
\[ \cos^2 A = 1 - \frac{1}{x^2} \]

5. **Combine the terms**:
To combine the terms on the right side, we can express 1 as \( \frac{x^2}{x^2} \):
\[ \cos^2 A = \frac{x^2}{x^2} - \frac{1}{x^2} = \frac{x^2 - 1}{x^2} \]

6. **Take the square root**:
To find \( \cos A \), we take the square root of both sides:
\[ \cos A = \sqrt{\frac{x^2 - 1}{x^2}} \]

7. **Simplify the expression**:
This can be simplified further:
\[ \cos A = \frac{\sqrt{x^2 - 1}}{x} \]

### Final Answer:
\[ \cos A = \frac{\sqrt{x^2 - 1}}{x} \]