Trigonometry & Inverse Trigonometry
Trigonometric identities
Grade 11
Question:
<p>The value of the expression <math>\cos^2\left(\frac{\pi}{8}\right) + \cos^2\left(\frac{3\pi}{8}\right) + \cos^2\left(\frac{5\pi}{8}\right) + \cos^2\left(\frac{7\pi}{8}\right)</math> is</p>
<p>(a) rational</p>
<p>(b) integral</p>
<p>(c) prime</p>
<p>(d) composite</p>
Step-by-Step Solution
Key Concept: Use complementary angle relationships and symmetry to simplify the sum of cosine squares
<p><strong>Step 1:</strong> Note that <math>\cos\left(\frac{7\pi}{8}\right) = -\cos\left(\frac{\pi}{8}\right)</math> and <math>\cos\left(\frac{5\pi}{8}\right) = -\cos\left(\frac{3\pi}{8}\right)</math></p><p><strong>Step 2:</strong> Therefore, <math>\cos^2\left(\frac{7\pi}{8}\right) = \cos^2\left(\frac{\pi}{8}\right)</math> and <math>\cos^2\left(\frac{5\pi}{8}\right) = \cos^2\left(\frac{3\pi}{8}\right)</math></p><p><strong>Step 3:</strong> The sum becomes <math>2\cos^2\left(\frac{\pi}{8}\right) + 2\cos^2\left(\frac{3\pi}{8}\right) = 2</math></p><p>∴ Answer is (b) integral</p>
Correct Answer: b