Question

If \( \csc\theta = 2x \) and \( \cot\theta = \frac{2}{x} \) then \( 2\left(x^2 - \frac{1}{x^2}\right) = ? \)

• A. 1
• B. 0
• C. \(\frac{1}{2}\)
• D. -1

Hint:

1. Use the identity: \( \csc^2\theta - \cot^2\theta = 1 \). 2. Substitute given values: \( \csc\theta = 2x \implies \csc^2\theta = (2x)^2 = 4x^2 \). \( \cot\theta = \frac{2}{x} \implies \cot^2\theta = \left(\frac{2}{x}\right)^2 = \frac{4}{x^2} \). 3. Plug into identity: \( 4x^2 - \frac{4}{x^2} = 1 \). 4. Factor out 4: \( 4\left(x^2 - \frac{1}{x^2}\right) = 1 \). 5. The question asks for \( 2\left(x^2 - \frac{1}{x^2}\right) \). Since \( 4\left(x^2 - \frac{1}{x^2}\right) = 1 \), then \( 2 \times 2\left(x^2 - \frac{1}{x^2}\right) = 1 \). So, \( 2\left(x^2 - \frac{1}{x^2}\right) = \frac{1}{2} \).

Answer 1

The Correct Option is C

Concept: This problem uses the fundamental trigonometric identity relating cosecant (\(\csc\)) and cotangent (\(\cot\)): \[ \csc^2\theta - \cot^2\theta = 1 \] Step 1: Substitute the given expressions for \(\csc\theta\) and \(\cot\theta\) into the identity Given: \(\csc\theta = 2x\) \(\cot\theta = \frac{2}{x}\) Substitute these into \( \csc^2\theta - \cot^2\theta = 1 \): \[ (2x)^2 - \left(\frac{2}{x}\right)^2 = 1 \] Step 2: Simplify the equation \[ (2x)^2 = 4x^2 \] \[ \left(\frac{2}{x}\right)^2 = \frac{2^2}{x^2} = \frac{4}{x^2} \] So the equation becomes: \[ 4x^2 - \frac{4}{x^2} = 1 \] Step 3: Factor out the common term to match the desired expression We need to find the value of \( 2\left(x^2 - \frac{1}{x^2}\right) \). Look at the equation from Step 2: \( 4x^2 - \frac{4}{x^2} = 1 \). Factor out 4 from the left side: \[ 4\left(x^2 - \frac{1}{x^2}\right) = 1 \] Step 4: Solve for the desired expression We want \( 2\left(x^2 - \frac{1}{x^2}\right) \). From \( 4\left(x^2 - \frac{1}{x^2}\right) = 1 \), we can find \( \left(x^2 - \frac{1}{x^2}\right) \) by dividing by 4: \[ x^2 - \frac{1}{x^2} = \frac{1}{4} \] Now, multiply by 2: \[ 2\left(x^2 - \frac{1}{x^2}\right) = 2 \times \frac{1}{4} \] \[ 2\left(x^2 - \frac{1}{x^2}\right) = \frac{2}{4} = \frac{1}{2} \] Thus, the value of the expression is \(\frac{1}{2}\).