If `cosec theta = 3x` and `cot theta` = 3/x, then what is the value of `x^2- 1//x^2`?

Options

A. A)3
B. B)`1//3`
C. C)9
D. D)`1//9`

Answer

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

1. \( \csc \theta = 3x \)
2. \( \cot \theta = \frac{3}{x} \)

We need to find the value of \( x^2 - \frac{1}{x^2} \).

### Step 1: Use the identity
We know from trigonometric identities that:
\[ \csc^2 \theta - \cot^2 \theta = 1 \]

### Step 2: Square the given values
Now, let's square both sides of the given equations:

- From \( \csc \theta = 3x \):
\[ \csc^2 \theta = (3x)^2 = 9x^2 \]

- From \( \cot \theta = \frac{3}{x} \):
\[ \cot^2 \theta = \left(\frac{3}{x}\right)^2 = \frac{9}{x^2} \]

### Step 3: Substitute into the identity
Now substitute these squared values into the identity:
\[ 9x^2 - \frac{9}{x^2} = 1 \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ 9x^2 - \frac{9}{x^2} = 1 \]
\[ 9x^2 = 1 + \frac{9}{x^2} \]

### Step 5: Multiply through by \( x^2 \)
To eliminate the fraction, multiply through by \( x^2 \):
\[ 9x^4 = x^2 + 9 \]

### Step 6: Rearranging to form a quadratic equation
Rearranging gives us:
\[ 9x^4 - x^2 - 9 = 0 \]

### Step 7: Let \( y = x^2 \)
Let \( y = x^2 \), then we have:
\[ 9y^2 - y - 9 = 0 \]

### Step 8: Solve the quadratic equation
Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 9 \), \( b = -1 \), and \( c = -9 \).
\[ y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 9 \cdot (-9)}}{2 \cdot 9} \]
\[ y = \frac{1 \pm \sqrt{1 + 324}}{18} \]
\[ y = \frac{1 \pm \sqrt{325}}{18} \]

### Step 9: Finding \( x^2 - \frac{1}{x^2} \)
Now we need to find \( x^2 - \frac{1}{x^2} \):
\[ x^2 - \frac{1}{x^2} = y - \frac{1}{y} \]
\[ = \frac{y^2 - 1}{y} \]

### Step 10: Substitute \( y \)
Now substituting \( y = \frac{1 + \sqrt{325}}{18} \) or \( y = \frac{1 - \sqrt{325}}{18} \) into \( y^2 - 1 \) and simplifying will yield the final answer.

However, we can also directly use the earlier derived equation:
\[ 9x^2 - \frac{9}{x^2} = 1 \implies x^2 - \frac{1}{x^2} = \frac{1}{9} \]

### Final Answer
Thus, the value of \( x^2 - \frac{1}{x^2} \) is:
\[ \frac{1}{9} \]