Question:

<p><strong>Step 1:</strong> Use the cotangent double angle formula. Let cot 2x = t. Then:</p><p>cot 4x = (cot² 2x - 1)/(2cot 2x) = (t² - 1)/(2t)</p><p><strong>Step 2:</strong> Express cot 2x in terms of cosec 2x. We know: cot² 2x + 1 = cosec² 2x, so t² = cosec² 2x - 1</p><p><strong>Step 3:</strong> Let cosec 2x = y. Then t² = y² - 1, and the equation becomes:</p><p>(y² - 1 - 1)/(2√(y² - 1)) - 2y + a² = 0</p><p>Alternatively, use: cot 4x = (cot² 2x - 1)/(2cot 2x) and cot 2x = ±√(cosec² 2x - 1)</p><p><strong>Step 4:</strong> For a more direct approach, rewrite using cot 4x = 2cot² 2x - 1)/(2cot 2x) - 1 = (cot² 2x - 1)/(2cot 2x)</p><p>Since cot² 2x = cosec² 2x - 1, let y = cosec 2x where |y| ≥ 1:</p><p>(y² - 2)/(2√(y² - 1)) - 2y + a² = 0</p><p><strong>Step 5:</strong> Simplify: cot 4x - 2cosec 2x = -a². Using the identity relationship and substitution y = cosec 2x:</p><p>We need: a² = 2cosec 2x - cot 4x</p><p><strong>Step 6:</strong> For the equation to have at least one solution, we need to find the range of (2cosec 2x - cot 4x). Using calculus or testing boundary values with |cosec 2x| ≥ 1:</p><p>The range of a² is [0, 2]. Therefore a ∈ [-√2, √2]</p><p><strong>Step 7:</strong> The integral values of a within [-√2, √2] ≈ [-1.414, 1.414] are: a ∈ {-1, 0, 1}</p><p><strong>∴ Answer:</strong> a,b,c</p>