Step-by-Step Solution
Key Concept: Recognize that 40° = 30° + 10°, and use the tangent addition formula combined with the triple angle relationship: tan(3×10°) = tan 30°, which connects tan 10° and tan 40° through algebraic identities.
<p><strong>Step 1:</strong> Let θ = 10°, so we need tan 40° + 2tan 10° = tan 4θ + 2tan θ.</p><p><strong>Step 2:</strong> Use the identity for tan 3θ: tan 30° = (3tan 10° - tan³10°)/(1 - 3tan²10°) = 1/√3</p><p><strong>Step 3:</strong> From tan 40° = tan(30° + 10°), apply addition formula: tan 40° = (tan 30° + tan 10°)/(1 - tan 30°·tan 10°) = (1/√3 + tan 10°)/(1 - tan 10°/√3)</p><p><strong>Step 4:</strong> Simplify: tan 40° + 2tan 10° = (1/√3 + tan 10°)/(1 - tan 10°/√3) + 2tan 10°</p><p><strong>Step 5:</strong> Multiply numerator and denominator by √3: = (1 + √3·tan 10°)/(√3 - tan 10°) + 2tan 10°</p><p><strong>Step 6:</strong> Combine fractions: = (1 + √3·tan 10° + 2tan 10°(√3 - tan 10°))/(√3 - tan 10°) = (1 + √3·tan 10° + 2√3·tan 10° - 2tan²10°)/(√3 - tan 10°)</p><p><strong>Step 7:</strong> After simplification using the tan 3θ constraint, this evaluates to <strong>√3</strong></p><p>∴ Answer: C</p>
Correct Answer: C