Trigonometry & Inverse Trigonometry
Trigonometric Identities
Grade 11
Question:
<p>The value of \(\cos^2\frac{\pi}{16}+\cos^2\frac{3\pi}{16}+\cos^2\frac{5\pi}{16}+\cos^2\frac{7\pi}{16}\) is</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) -1</p>
<p>(d) 2</p>
Step-by-Step Solution
Key Concept: Recognize complementary angle pairs and use the identity $\sin^2\theta + \cos^2\theta = 1$ to pair and simplify the sum.
<p><strong>Step 1:</strong> Use the complementary angle relationship.</p><p><strong>Step 2:</strong> Note that $\frac{5\pi}{16} = \frac{\pi}{2} - \frac{3\pi}{16}$ and $\frac{7\pi}{16} = \frac{\pi}{2} - \frac{\pi}{16}$</p><p><strong>Step 3:</strong> Therefore:</p><p>$\cos^2\frac{5\pi}{16} = \cos^2\left(\frac{\pi}{2}-\frac{3\pi}{16}\right) = \sin^2\frac{3\pi}{16}$</p><p>$\cos^2\frac{7\pi}{16} = \cos^2\left(\frac{\pi}{2}-\frac{\pi}{16}\right) = \sin^2\frac{\pi}{16}$</p><p><strong>Step 4:</strong> $\left(\cos^2\frac{\pi}{16}+\sin^2\frac{\pi}{16}\right) + \left(\cos^2\frac{3\pi}{16}+\sin^2\frac{3\pi}{16}\right) = 1 + 1 = 2$</p><p>∴ Answer is (d) 2</p>
Correct Answer: D