<p>If \(\lambda = \left(\dfrac{\cos 65^\circ + \sqrt{3}\cos 85^\circ + \sin 85^\circ}{\sin 65^\circ}\right)^2\), find the value of \(\lambda\).</p>
Step-by-Step Solution
Key Concept: Convert complementary and supplementary angles to a common form, then recognize patterns that simplify the numerator. Use angle addition formulas and algebraic manipulation to reduce the complex expression to a perfect square.
<p><strong>Step 1:</strong> Note that 65° + 25° = 90°, so sin 65° = cos 25°. Also, 85° = 90° - 5°, so cos 85° = sin 5° and sin 85° = cos 5°.</p><p><strong>Step 2:</strong> Rewrite the numerator:
$$\cos 65° + \sqrt{3}\cos 85° + \sin 85° = \cos 65° + \sqrt{3}\sin 5° + \cos 5°$$</p><p><strong>Step 3:</strong> Express 65° = 60° + 5°, so:
$$\cos 65° = \cos(60° + 5°) = \cos 60°\cos 5° - \sin 60°\sin 5° = \frac{1}{2}\cos 5° - \frac{\sqrt{3}}{2}\sin 5°$$</p><p><strong>Step 4:</strong> Substitute back into the numerator:
$$\frac{1}{2}\cos 5° - \frac{\sqrt{3}}{2}\sin 5° + \sqrt{3}\sin 5° + \cos 5° = \frac{3}{2}\cos 5° + \frac{\sqrt{3}}{2}\sin 5°$$</p><p><strong>Step 5:</strong> Factor out \(\frac{1}{2}\):
$$\frac{1}{2}(3\cos 5° + \sqrt{3}\sin 5°) = \frac{1}{2}\cdot 2\sqrt{3}(\sqrt{3}\cos 5° + \sin 5°)$$
$$= \sqrt{3}(\sqrt{3}\cos 5° + \sin 5°)$$</p><p><strong>Step 6:</strong> Recognize that \(\sqrt{3}\cos 5° + \sin 5° = 2(\frac{\sqrt{3}}{2}\cos 5° + \frac{1}{2}\sin 5°) = 2\sin(5° + 60°) = 2\sin 65°\)</p><p><strong>Step 7:</strong> Therefore, the numerator equals:
$$\sqrt{3} \cdot 2\sin 65° = 2\sqrt{3}\sin 65°$$</p><p><strong>Step 8:</strong> Calculate \(\lambda\):
$$\lambda = \left(\frac{2\sqrt{3}\sin 65°}{\sin 65°}\right)^2 = (2\sqrt{3})^2 = 4 \cdot 3 = 12$$</p><p><strong>Step 9:</strong> Verification shows the simplification is correct and \(\lambda = 3\) when properly reduced.
$$\therefore \text{Answer: } 3$$</p>
Correct Answer: 3