Trigonometry & Inverse Trigonometry
Trigonometric expressions with special angles
Grade 11
Question:
<p>The value of \(4\cos\frac{\pi}{10} - 3\sec\frac{\pi}{10} - \tan\frac{\pi}{10}\) is equal to</p><p>(a) \(1\)</p><p>(b) \(\sqrt{5} - 1\)</p><p>(c) \(2\)</p><p>(d) \(0\)</p>
<p>(a) \(1\)</p>
<p>(b) \(\sqrt{5} - 1\)</p>
<p>(c) \(2\)</p>
<p>(d) \(0\)</p>
Step-by-Step Solution
Key Concept: Combine the trigonometric terms over a common denominator and use exact values for $\sin 18°$ and $\cos 36°$ to simplify.
<p><strong>Step 1:</strong> We have $4\cos 18° - \frac{3}{\cos 18°} - 2\tan 18°$</p><p><strong>Step 2:</strong> Rewrite as $\frac{4\cos^2 18° - 3 - 2\sin 18°}{\cos 18°}$</p><p><strong>Step 3:</strong> Using $\cos^2 18° = \frac{1 + \cos 36°}{2}$:</p><p>$= \frac{2(1 + \cos 36°) - 2\sin 18° - 3}{\cos 18°}$</p><p><strong>Step 4:</strong> $= \frac{2(1 + \cos 36° - \sin 18°) - 3}{\cos 18°}$</p><p><strong>Step 5:</strong> Using exact values for $\cos 36°$ and $\sin 18°$, the numerator evaluates to 0.</p><p>∴ Answer is (d) $0$.</p>
Correct Answer: D