Final answer:
The question concerns rewriting sin(x * π/6) * sin(x * π/6) using sin(x) and cos(x), but there is not enough information or a direct trigonometric identity to complete the transformation. None of the provided options A through D correctly rewrite the expression using only sin(x) and cos(x).
Step-by-step explanation:
The student's question asks to rewrite the expression sin(x * π/6) * sin(x * π/6) in terms of sin(x) and cos(x). To do this, we should first note that subtracting a function's argument from π is equivalent to multiplying the function by -1. However, since we're squaring the result, the negative sign would be eliminated. Using trigonometric identities, we know that sin(π - θ) = sin(θ) and cos(π - θ) = -cos(θ). Therefore, we can rewrite sin(x * π/6) as sin(π/6 - (x * π/6) - π/6), which simplifies to sin((1-x) * π/6).
Even with this simplification, there is not a straightforward identity to directly rewrite the expression with sin(x) and cos(x) without additional information or context. The provided identities and options A) sin²(x) * cos²(x), B) 1/2 * (1 - cos(2x)), C) 1/2 * (1 + cos(2x)), and D) cos²(x) - sin²(x) are not directly applicable for rewriting sin(x * π/6) * sin(x * π/6). If we were attempting to express sin²(π/6) in terms of sin(x) and cos(x), we would need additional steps that involve specific angle values or further identities.
Given the context of the problem and the trigonometric identities available, we cannot solve the mathematical problem completely without further assumptions or information about the value of x. Thus, none of the provided options A through D correctly represent the expression sin(x * π/6) * sin(x * π/6) in terms of sin(x) * sin(x) and cos(x) * cos(x).