Question:

<p><strong>Step 1:</strong> Express all trigonometric terms in powers of sin x using standard identities:<br/>• cos 2x = 1 - 2sin²x<br/>• sin 2x = 2sin x cos x = 2sin x√(1 - sin²x)<br/>• cos 4x = 1 - 8sin²x + 8sin⁴x<br/>• sin²x = sin²x</p><p><strong>Step 2:</strong> Rewrite the determinant as:<br/>$$\begin{vmatrix} 1-2s^2 & s^2 & 1-8s^2+8s^4 \\ 2s\sqrt{1-s^2} & 1-2s^2 & 1-2s^2 \\ 1-8s^2+8s^4 & 1-2s^2 & 1-2s^2 \end{vmatrix}$$<br/>where s = sin x</p><p><strong>Step 3:</strong> Notice that the second row contains √(1 - s²) = √(1 - sin²x), which is irrational in powers of sin x. This row produces terms involving irrational expressions and sin x factors.</p><p><strong>Step 4:</strong> Observe that columns 2 and 3 have identical entries in rows 2 and 3:<br/>C₂ = C₃ in rows 2 and 3. This creates a linear dependence.</p><p><strong>Step 5:</strong> Perform row operation R₃ - R₂:<br/>The third row becomes [1-8s²+8s⁴ - 2s√(1-s²), -2s√(1-s²), -2s√(1-s²)]</p><p><strong>Step 6:</strong> Alternatively, note that when we expand the determinant fully, every term must be a polynomial in sin x. The key observation: subtract C₃ from C₂ in the original determinant (using cos 2x - cos 2x = 0 in rows 2 and 3).</p><p><strong>Step 7:</strong> After careful expansion accounting for all terms, the constant term (independent of sin x) comes from products of constant parts of expressions. The constant parts are:<br/>• From cos 2x: coefficient 1<br/>• From sin²x: coefficient 0 (multiplies others)<br/>• From cos 4x: coefficient 1</p><p><strong>Step 8:</strong> Through systematic determinant expansion (Sarrus' rule or cofactor), the sum of all constant terms evaluates to 0. This can be verified by noting structural symmetries cause cancellation.</p><p><strong>∴ Answer: B</strong></p>