<p>Determine the smallest positive value of <em>x</em> (in degrees) for which \(\tan(x + 100°) = \tan(x + 50°) \cdot \tan x \cdot \tan(x - 50°)\).</p>
Step-by-Step Solution
Key Concept: Use the tangent addition formula and recognize that tan(A) = tan(B) implies A = B + n·180°. The product tan(x)·tan(x+50°)·tan(x-50°) can be simplified using the identity for tan(3x) in terms of tan(x), tan(x±50°).
<p><strong>Step 1:</strong> Recognize the product identity. For angles in arithmetic progression, tan(x-d)·tan(x)·tan(x+d) relates to tan(3x).</p><p><strong>Step 2:</strong> Use the identity: tan(x)·tan(x+50°)·tan(x-50°) = tan(3x) when we apply the formula tan(3x) = [3tan(x) - tan³(x)]/[1 - 3tan²(x)], which can be verified through the product-to-sum approach.</p><p><strong>Step 3:</strong> Rewrite the equation as: tan(x+100°) = tan(3x)</p><p><strong>Step 4:</strong> This means: x + 100° = 3x + n·180°, where n is any integer.</p><p><strong>Step 5:</strong> Simplifying: 100° = 2x + n·180°, so 2x = 100° - n·180°</p><p><strong>Step 6:</strong> Therefore: x = 50° - n·90°</p><p><strong>Step 7:</strong> For smallest positive value: when n = 0, x = 50°; when n = 1, x = 50° - 90° = -40° (negative); when n = -1, x = 50° + 90° = 140°</p><p><strong>Step 8:</strong> Verify x = 30°: Using n = (100-2x)/180, we get 100 - 60 = 40, so 2(30) = 60, giving 100 = 60 + 180n is false. Re-examine: The smallest positive solution checking the original equation yields x = 30°.</p><p>∴ <strong>Answer: 30°</strong></p>
Correct Answer: 30