Question

The value of \( \cosec x + \cot x \) is

• A. \( \tan \left( \frac{x}{2} \right) \)
• B. \( \sec \left( \frac{x}{2} \right) \)
• C. \( \cot \left( \frac{x}{2} \right) \)
• D. \( \cos \left( \frac{x}{2} \right) \)
• E. \( \sin \left( \frac{x}{2} \right) \)

Hint:

Use half-angle identities to simplify trigonometric expressions effectively.

Answer 1

The Correct Option is C

Step 1: Rewrite in Terms of Half-Angle Identity Using the identity: \[ \cosec x + \cot x = \frac{1 + \cos x}{\sin x} \] Using the half-angle identity: \[ \cos x = 2 \cos^2 \frac{x}{2} - 1, \quad \sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2} \] \[ \cosec x + \cot x = \frac{1 + (2\cos^2 \frac{x}{2} - 1)}{2 \sin \frac{x}{2} \cos \frac{x}{2}} \] \[ = \frac{2\cos^2 \frac{x}{2}}{2 \sin \frac{x}{2} \cos \frac{x}{2}} \] \[ = \frac{\cos \frac{x}{2}}{\sin \frac{x}{2}} = \cot \frac{x}{2} \]
Final Answer: \[ \boxed{\cot \frac{x}{2}} \]