Find the value of the sum log (tan 10°) + log (tan 20°) + log (tan 30°) + … + log (tan 80°), where all logarithms are to base 10.

Introduction / Context:
This trigonometric logarithm problem asks you to evaluate a sum of logarithms of tangent values at specific angles. The angles come in complementary pairs, and the key insight is to use the identity involving tan θ and tan(90° − θ). Understanding this symmetry helps dramatically simplify the expression without computing any actual tangent values.

Given Data / Assumptions:

• The sum is S = log(tan 10°) + log(tan 20°) + log(tan 30°) + log(tan 40°) + log(tan 50°) + log(tan 60°) + log(tan 70°) + log(tan 80°).
• All logarithms are base 10.
• Angles are in degrees, and 0° < angle < 90°.

Concept / Approach:
We make use of the identity tan(90° − θ) = cot θ. The product tan θ × tan(90° − θ) = 1, since tan θ × cot θ = 1. Taking logarithms, log(tan θ) + log(tan(90° − θ)) = log 1 = 0. The given angles can be grouped into complementary pairs: (10°, 80°), (20°, 70°), (30°, 60°), and (40°, 50°). Each pair contributes zero to the sum of logs, so the entire sum collapses to zero.

Step-by-Step Solution:
Recall that tan(90° − θ) = cot θ and tan θ × cot θ = 1. Therefore tan θ × tan(90° − θ) = 1 for 0° < θ < 90°. Taking logarithms, log(tan θ) + log(tan(90° − θ)) = log 1 = 0. Now group the terms in S into complementary pairs: Pair 1: log(tan 10°) + log(tan 80°) = 0. Pair 2: log(tan 20°) + log(tan 70°) = 0. Pair 3: log(tan 30°) + log(tan 60°) = 0. Pair 4: log(tan 40°) + log(tan 50°) = 0. Add all pairs: S = 0 + 0 + 0 + 0 = 0.

Verification / Alternative check:
If desired, you can approximate a couple of pairs numerically. For example, tan 30° = 1/√3 and tan 60° = √3, so tan 30° × tan 60° = 1. Therefore log(tan 30°) + log(tan 60°) = log 1 = 0. Similar behaviour holds for (10°, 80°) and other pairs, confirming that every complementary pair contributes zero to the sum.

Why Other Options Are Wrong:
-1, 1/2, and 1: All these would imply that the product of all the tangent values is something other than 1, but the pairing argument shows that each pair multiplies to 1, and hence the overall product is 1 and the sum of logs must be 0.

Common Pitfalls:
A typical mistake is attempting to compute or approximate each tangent value directly, which is unnecessary and time consuming. Another pitfall is forgetting the identity tan(90° − θ) = cot θ, which is central to the simplification. Also, some students mis-pair angles or assume an incorrect period for tangent, leading to sign mistakes.

Final Answer:
The value of the sum of logarithms is 0.