The solution is found by simplifying the given trigonometric expression and using fundamental identities. Sine of x must equal 1 for the equation to hold, which occurs at x = π/2, the only provided option where this is true.
The equation given is cos(x).tan(x)/csc²(x) = sin³(x). To solve this, you can use trigonometric identities. csc(x) is equal to 1/sin(x), tan(x) is equal to sin(x)/cos(x), so that cos(x) * (sin(x)/cos(x)) becomes just sin(x).
The equation simplifies to sin(x)/sin²(x) = sin³(x).
After simplification, you get 1/sin(x) = sin²(x), which further simplifies to 1 = sin³(x) * sin(x).
This means that sin(x) must equal 1 or -1 for the equation to hold true. By looking at the unit circle, we can see that sin(x) equals 1 at x = π/2, and -1 at x = 3π/2. However, only x = π/2 is an option from the given answers.