If `cos^(-1)" 1/x=theta , " then " tan theta=`

Options

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

Answer

To solve the problem, we start with the equation given:

1. **Given**: \( \cos^{-1}\left(\frac{1}{x}\right) = \theta \)

2. **Step 1**: Rewrite the equation in terms of cosine:
\[ \frac{1}{x} = \cos(\theta) \]

3. **Step 2**: Take the reciprocal of both sides:
\[ x = \sec(\theta) \]

4. **Step 3**: Use the identity relating secant and tangent:
\[ \sec^2(\theta) = 1 + \tan^2(\theta) \]

5. **Step 4**: Substitute \( x \) into the identity:
\[ x^2 = \sec^2(\theta) = 1 + \tan^2(\theta) \]

6. **Step 5**: Rearranging the equation to find \( \tan^2(\theta) \):
\[ \tan^2(\theta) = x^2 - 1 \]

7. **Step 6**: Taking the square root to find \( \tan(\theta) \):
\[ \tan(\theta) = \sqrt{x^2 - 1} \]

Thus, the final result is:
\[ \tan(\theta) = \sqrt{x^2 - 1} \]