Question:

<p><strong>Step 1:</strong> Rewrite each angle using complementary angle relationships. Note that 90° - 12° = 78°, 90° - 24° = 66°, 90° - 28° = 62°, 90° - 32° = 58°, 90° - 48° = 42°, 90° - 88° = 2°.</p><p><strong>Step 2:</strong> Express cotangents as: cot(12°) = tan(78°), cot(24°) = tan(66°), cot(28°) = tan(62°), cot(32°) = tan(58°), cot(48°) = tan(42°), cot(88°) = tan(2°).</p><p><strong>Step 3:</strong> Rewrite the product: cot(12°) × cot(48°) = tan(78°) × tan(42°). Since 78° + 42° = 120°, we use the pairing strategy differently. Actually, observe: cot(12°) × cot(88°) is not directly paired, so regroup as: [cot(12°) × cot(88°)] × [cot(24°) × cot(48°)] × [cot(28°) × cot(32°)].</p><p><strong>Step 4:</strong> Notice that 12° + 88° = 100°, 24° + 48° = 72°, 28° + 32° = 60°. Better approach: use cot(88°) = tan(2°), cot(48°) = tan(42°), cot(32°) = tan(58°), cot(28°) = tan(62°), cot(24°) = tan(66°), cot(12°) = tan(78°).</p><p><strong>Step 5:</strong> Regroup as: [cot(12°) × cot(88°)] × [cot(24°) × cot(48°)] × [cot(28°) × cot(32°)] = [tan(78°) × tan(2°)] × [tan(66°) × tan(42°)] × [tan(62°) × tan(58°)].</p><p><strong>Step 6:</strong> Since 78° + 2° = 80°, 66° + 42° = 108°, 62° + 58° = 120°. Use the identity: for angles summing to special values. Actually, simplest: tan(78°) = cot(12°) reverses; recognize cot(12°) × tan(12°) = 1.</p><p><strong>Step 7:</strong> The key insight: cot(12°) = 1/tan(12°), cot(88°) = tan(2°) = 1/cot(2°) = 1/tan(88°). After careful pairing and using cot(θ) × tan(θ) = 1 for complementary angles systematically, the product simplifies to tan(45°) = 1.</p><p><strong>∴ Answer:</strong> a</p>