JEEOlympiad
Evaluate the product P = (1 + tan 1°)(1 + tan 2°)(1 + tan 3°) ⋯ (1 +…
Question
Evaluate the product P = (1 + tan 1°)(1 + tan 2°)(1 + tan 3°) ⋯ (1 + tan 44°). Report the value of P.
✓ Verified answer: 4194304checked by our engine — not a guess
Step-by-step solution
Key identity: for angles summing to 45°, (1 + tan A)(1 + tan B) = 2 when A + B = 45°. Proof: tan(A+B) = tan 45° = 1 = (tan A + tan B)/(1 − tan A tan B), so tan A + tan B = 1 − tan A tan B, hence (1 + tan A)(1 + tan B) = 1 + tan A + tan B + tan A tan B = 1 + (1 − tan A tan B) + tan A tan B = 2.
Pair the factors of P as (1°, 44°), (2°, 43°), …, (22°, 23°). Each pair sums to 45° and multiplies to 2. There are 22 such pairs covering all 44 factors.
Thus P = 2²² = 4194304.
Final answer4194304
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