<p>\(\sin 47° + \sin 61° - \sin 11° - \sin 25°\) is equal to</p>
Step-by-Step Solution
Key Concept: Use sum-to-product formulas to convert sums of sines into products, then factor out common terms.
<p><strong>Step 1:</strong> Group the terms strategically: $(\sin 47° + \sin 61°) - (\sin 11° + \sin 25°)$</p><p><strong>Step 2:</strong> Apply sum-to-product formula $\sin C + \sin D = 2\sin\frac{C+D}{2}\cos\frac{C-D}{2}$</p><p><strong>Step 3:</strong> $\sin 47° + \sin 61° = 2\sin 54°\cos 7°$</p><p><strong>Step 4:</strong> $\sin 11° + \sin 25° = 2\sin 18°\cos 7°$</p><p><strong>Step 5:</strong> Expression becomes $2\cos 7°(\sin 54° - \sin 18°)$</p><p><strong>Step 6:</strong> Note that $\sin 54° = \cos 36°$ and $\sin 18° = \frac{\sqrt{5}-1}{4}$, and using $\sin 54° - \sin 18° = \frac{1}{2}$</p><p><strong>Step 7:</strong> Result simplifies to $\cos 7°$</p><p>∴ Answer is (b) $\cos 7°$</p>
Correct Answer: B