Find the value of log10 (tan 10°) + log10 (tan 20°) + log10 (tan 30°) + log10 (tan 40°) + log10 (tan 50°) + log10 (tan 60°) + log10 (tan 70°) + log10 (tan 80°).

Introduction / Context:
This question combines trigonometry and logarithms. It asks for the value of a sum of logarithms of tangent values at various angles. Instead of computing each tangent and each logarithm separately, which would be tedious and inaccurate, we should use trigonometric identities and log properties to simplify the entire expression. The key idea is the complementary angle identity for tangent, which leads to products equal to 1 and hence logarithms equal to 0.

Given Data / Assumptions:
- We must evaluate S = log10 (tan 10°) + log10 (tan 20°) + log10 (tan 30°) + log10 (tan 40°) + log10 (tan 50°) + log10 (tan 60°) + log10 (tan 70°) + log10 (tan 80°).
- All logarithms are common logarithms with base 10.
- The tangent function is defined for these angles in degrees, and none of these angles is an odd multiple of 90°, so tan values are finite.

Concept / Approach:
The main trigonometric identity we use is tan θ * tan (90° - θ) = 1 for angles where both tangent values are defined. Taking logarithms of both sides gives log10 (tan θ) + log10 (tan (90° - θ)) = log10 1 = 0. This means that each pair of complementary angles contributes zero to the total sum of logs. The given angles naturally form such complementary pairs: 10° with 80°, 20° with 70°, 30° with 60°, and 40° with 50°.

Step-by-Step Solution:
Observe the complementary angle pairs: (10°, 80°), (20°, 70°), (30°, 60°), (40°, 50°). Use the identity tan θ * tan (90° - θ) = 1 for each pair. For θ = 10°, tan 10° * tan 80° = 1. Taking common logs: log10 (tan 10°) + log10 (tan 80°) = log10 1 = 0. Similarly, log10 (tan 20°) + log10 (tan 70°) = 0. Also, log10 (tan 30°) + log10 (tan 60°) = 0. And log10 (tan 40°) + log10 (tan 50°) = 0. Now group the original sum S into these four pairs. S = [log10 (tan 10°) + log10 (tan 80°)] + [log10 (tan 20°) + log10 (tan 70°)] + [log10 (tan 30°) + log10 (tan 60°)] + [log10 (tan 40°) + log10 (tan 50°)]. Each bracket equals 0, so S = 0 + 0 + 0 + 0 = 0.

Verification / Alternative check:
As a numerical check, we can approximate a few values. For example, tan 30° is 1 / sqrt(3) and tan 60° is sqrt(3). Their product is 1, so log10 (tan 30°) + log10 (tan 60°) = log10 1 = 0, matching the theoretical identity. Likewise, tan 45° is 1, and although 45° does not appear here, it serves as a reminder that tangent and its complementary values often form simple pairs. Using a calculator for a few pairs confirms that each pair of logs sums to zero within rounding error.

Why Other Options Are Wrong:
-1: This would require the overall product of all tan values to be 10^-1, which contradicts the pair wise product being 1 for each complementary pair.
1/2 or 1: These non zero values would imply that the product of all tan values is a power of 10 different from 1, again contradicting the identity tan θ * tan (90° - θ) = 1 for each pair.

Common Pitfalls:
Students may overlook the complementary angle pattern and attempt to compute each term individually, which is slow and error prone. Another common mistake is to apply the sine or cosine complementary identities instead of the tangent identity, or to incorrectly think that tan (90° - θ) equals cot θ without using the full product identity. Forgetting that log10 1 equals 0 can also lead to confusion. Recognizing symmetry in angle sets and pairing complementary angles is the key to these problems.

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