Question

Sham is trying to solve the expression:
\[ \log \tan 1^\circ + \log \tan 2^\circ + \log \tan 3^\circ + \ldots + \log \tan 89^\circ. \]  
The correct answer would be?

• A. 1
• B. \(\frac{1}{\sqrt{2}}\)
• C. 0
• D. -1

Answer 1

The Correct Option is C

Step 1: Let \(S=\sum_{k=1}^{89}\log\tan k^{\circ}\). Replacing \(k\) with \(90-k\) gives the same sum written as \(\sum \log\cot k^{\circ}\).
Step 2: Adding both versions: \(2S=\sum\log(\tan k^{\circ}\cot k^{\circ})=\sum\log1=0\).
Step 3: So \(S=0\).
\[ \boxed{0} \]