Question

If \( \csc\theta - \cot\theta = 2 \), then the value of \( \csc\theta \) is

• A. \( \frac{5}{4} \)
• B. \( \frac{3}{4} \)
• C. \( \frac{1}{2} \)
• D. \( \frac{5}{2} \)

Hint:

Whenever you see \( \csc\theta \pm \cot\theta \) or \( \sec\theta \pm \tan\theta \), immediately use the Pythagorean identities \( \csc^2\theta - \cot^2\theta = 1 \) or \( \sec^2\theta - \tan^2\theta = 1 \). This allows you to find the value of the conjugate expression and solve the resulting system of equations.

Answer 1

The Correct Option is A

We are given the equation: \[ \csc\theta - \cot\theta = 2 \cdots(1) \] We know the fundamental trigonometric identity: \[ \csc^2\theta - \cot^2\theta = 1 \] Factoring the difference of squares: \[ (\csc\theta - \cot\theta)(\csc\theta + \cot\theta) = 1 \] Substitute the value from equation (1) into this identity: \[ (2)(\csc\theta + \cot\theta) = 1 \] \[ \csc\theta + \cot\theta = \frac{1}{2} \cdots(2) \] Now we have a system of two linear equations in \( \csc\theta \) and \( \cot\theta \). Adding equation (1) and equation (2): \[ (\csc\theta - \cot\theta) + (\csc\theta + \cot\theta) = 2 + \frac{1}{2} \] \[ 2\csc\theta = \frac{5}{2} \] \[ \csc\theta = \frac{5}{4} \] The OCR answer is D, which is incorrect. Let's recheck the calculation. \( 2\csc\theta = 2.5 \), so \( \csc\theta = 1.25 = 5/4 \). The answer is indeed 5/4, which is option A. The provided answer key (Ans: d) is incorrect.