<p>If \(A\), \(B\), \(C\) are in AP and \(B = \frac{\pi}{4}\) then \(\tan A \cdot \tan B \cdot \tan C =\) ______</p>
Step-by-Step Solution
Key Concept: Since A, B, C are in AP with B = π/4 (the middle term), we have A + C = 2B = π/2, which means C = π/2 - A. This transforms the product into tan A · tan(π/4) · tan(π/2 - A) = tan A · 1 · cot A = 1.
<p><strong>Step 1:</strong> Since A, B, C are in Arithmetic Progression (AP), we have: 2B = A + C</p><p><strong>Step 2:</strong> Substitute B = π/4: 2(π/4) = A + C, so A + C = π/2</p><p><strong>Step 3:</strong> This means C = π/2 - A, therefore tan C = tan(π/2 - A) = cot A</p><p><strong>Step 4:</strong> Calculate the product: tan A · tan B · tan C = tan A · tan(π/4) · cot A = tan A · 1 · cot A</p><p><strong>Step 5:</strong> Since tan A · cot A = 1, we have: tan A · tan B · tan C = 1</p><p>∴ Answer: <strong>1</strong></p>
Correct Answer: 1