Trigonometry & Inverse Trigonometry
Trigonometric identities and series
Grade 11
Question:
<p><strong>Ex. 14:</strong> If \(\csc \frac{7\pi}{32} + \csc \frac{7\pi}{16} + \csc \frac{7\pi}{8} + \csc \frac{7\pi}{4} = \csc \frac{7\pi}{2} - \cot \frac{7\pi}{k}\), then the value of k is</p>
<p>(a) 64</p>
<p>(b) 96</p>
<p>(c) 48</p>
<p>(d) 32</p>
Step-by-Step Solution
Key Concept: Recognize the telescoping series pattern where $\csc \theta = \cot \frac{\theta}{2} - \cot \theta$, allowing cancellation of intermediate terms.
<p><strong>Solution:</strong> Let $\theta = \frac{7\pi}{32}$</p><p>Then $T_n = \csc \theta - \cot n\theta$ where each term follows the telescoping pattern:</p><p>$T_1 = \csc \theta = \cot \frac{\theta}{2} - \cot \theta$</p><p>$T_2 = \cot \theta - \cot 2\theta$</p><p>$T_3 = \cot 2\theta - \cot 4\theta$</p><p>Summing up the telescoping series from the given equation and comparing both sides, we find that $k = 64$.</p>
Correct Answer: A