Final answer:
To simplify the expression cos(π/7) * cos(x) * sin(π/7) * sin(x), we can apply the trigonometric identity sin(a) * cos(b) = (1/2) * [sin(a + b) + sin(a - b)]. Expanding and simplifying the expression leads to option d) cos(2x - π/7).
Step-by-step explanation:
To simplify the expression cos(π/7) * cos(x) * sin(π/7) * sin(x), we can use the trigonometric identity sin(a) * cos(b) = (1/2) * [sin(a + b) + sin(a - b)].
Applying this identity twice, we have:
cos(π/7) * cos(x) * sin(π/7) * sin(x) = (1/2) * [sin(π/7 + x) + sin(π/7 - x)] * (1/2) * [sin(π/7 + x) - sin(π/7 - x)]
Expanding and simplifying further, we get:
cos(π/7) * cos(x) * sin(π/7) * sin(x) = (1/4) * [sin(2x + π/7) - sin(π/7 - 2x)]
Therefore, the simplified form of the expression is option d) cos(2x - π/7).