Question

The general solution of \( \cot \frac{x}{2} - \cot x = \csc \frac{x}{2} \) is:

• A. \( \{ 2n\pi + \frac{\pi}{3} | n \in \mathbb{Z} \} \)
• B. \( \{ 4n\pi + \frac{\pi}{3} | n \in \mathbb{Z} \} \)
• C. \( \{ 2n\pi + \frac{2\pi}{3} | n \in \mathbb{Z} \} \)
• D. \( \{ 4n\pi \pm \frac{2\pi}{3} | n \in \mathbb{Z} \} \)

Hint:

When solving trigonometric equations, remember to use identities and general solutions for the functions involved.

Answer 1

The Correct Option is D

To solve the equation: \[ \cot \frac{x}{2} - \cot x = \csc \frac{x}{2}, \] we proceed as follows: 
Step 1: Express \( \cot x \) in terms of \( \cot \frac{x}{2} \) Using the double-angle identity for cotangent: \[ \cot x = \frac{\cot^2 \frac{x}{2} - 1}{2 \cot \frac{x}{2}}. \] Substitute this into the equation: \[ \cot \frac{x}{2} - \frac{\cot^2 \frac{x}{2} - 1}{2 \cot \frac{x}{2}} = \csc \frac{x}{2}. \] Step 2: Simplify the equation
Multiply through by \( 2 \cot \frac{x}{2} \) to eliminate the denominator: \[ 2 \cot^2 \frac{x}{2} - (\cot^2 \frac{x}{2} - 1) = 2 \cot \frac{x}{2} \csc \frac{x}{2}. \] Simplify: \[ 2 \cot^2 \frac{x}{2} - \cot^2 \frac{x}{2} + 1 = 2 \cot \frac{x}{2} \csc \frac{x}{2}. \] Combine like terms: \[ \cot^2 \frac{x}{2} + 1 = 2 \cot \frac{x}{2} \csc \frac{x}{2}. \] Step 3: Use trigonometric identities Recall that \( \cot^2 \theta + 1 = \csc^2 \theta \). Thus: \[ \csc^2 \frac{x}{2} = 2 \cot \frac{x}{2} \csc \frac{x}{2}. \] Divide both sides by \( \csc \frac{x}{2} \): \[ \csc \frac{x}{2} = 2 \cot \frac{x}{2}. \] Rewrite \( \csc \frac{x}{2} \) and \( \cot \frac{x}{2} \) in terms of sine and cosine: \[ \frac{1}{\sin \frac{x}{2}} = 2 \cdot \frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}. \] Multiply through by \( \sin \frac{x}{2} \): \[ 1 = 2 \cos \frac{x}{2}. \] Step 4: Solve for \( \frac{x}{2} \) The equation simplifies to: \[ \cos \frac{x}{2} = \frac{1}{2}. \] The general solution for \( \cos \theta = \frac{1}{2} \) is: \[ \theta = 2n\pi \pm \frac{\pi}{3}, \quad n \in \mathbb{Z}. \] Thus, for \( \frac{x}{2} \): \[ \frac{x}{2} = 2n\pi \pm \frac{\pi}{3}. \] Multiply through by 2: \[ x = 4n\pi \pm \frac{2\pi}{3}, \quad n \in \mathbb{Z}. \] Final Answer: \[ \boxed{\{ 4n\pi \pm \frac{2\pi}{3} | n \in \mathbb{Z} \}} \]