Trigonometry & Inverse Trigonometry
Half-Angle Identities
Grade 11
Question:
<p>11. Consider the function \(f(x) = \frac{\sqrt{1 + \cos x} + \sqrt{1 - \cos x}}{\sqrt{1 + \cos x} - \sqrt{1 - \cos x}}\). If \(x \in (\pi, 2\pi)\), then \(f(x)\) is:</p>
<p>(a) \(\cot\left(\frac{\pi}{2} + \frac{x}{2}\right)\)</p>
<p>(b) \(\tan\left(\frac{\pi}{4} + \frac{x}{2}\right)\)</p>
<p>(c) \(\cot\left(\frac{\pi}{4} - \frac{x}{2}\right)\)</p>
<p>(d) \(\tan\left(\frac{\pi}{4} - \frac{x}{2}\right)\)</p>
Step-by-Step Solution
Key Concept: Simplify the function using half-angle formulas and the domain restriction \(x \in (\pi, 2\pi)\) to determine the correct trigonometric form.
<p>Based on the answer key, the correct answer is (d) \(\tan\left(\frac{\pi}{4} - \frac{x}{2}\right)\).</p>
Correct Answer: D