log(tan 2 degrees) + log(tan 4 degrees) + ... + log(tan 88 degrees) = ?

A. 1
B. 1/sqrt(2)
C. 0
D. -1

Question

log(tan 2 degrees) + log(tan 4 degrees) + ... + log(tan 88 degrees) = ?

A. 1
B. 1/sqrt(2)
C. 0
D. -1

Accepted Answer

S = 0 The key is the complementary angle identity: tan(theta) * tan(90 - theta) = 1, so log(tan theta) + log(tan(90 - theta)) = log(1) = 0. The 44 terms from 2 to 88 (step 2) pair as (2, 88), (4, 86), ..., (44, 46) -- exactly 22 pairs, each contributing 0 to the sum. Therefore the entire sum equals 0.

log(tan 2 degrees) + log(tan 4 degrees) + ... + log(tan 88 degrees) = ? A. 1 B. 1/sqrt(2) C. 0 D. -1