Find all real numbers in the interval (0, 2 pi) tanxsin2x=0
To find all real numbers in the interval (0, 2 pi) where tan(x)sin(2x) = 0, we need to find all solutions where either tan(x) = 0 or sin(2x) = 0.
Case 1: tan(x) = 0
Since tan(x) = sin(x)/cos(x), if tan(x) = 0, then sin(x) = 0. Thus, the solutions are x = 0 and x = pi.
Case 2: sin(2x) = 0
To solve sin(2x) = 0, we can rewrite it as 2sin(x)cos(x) = 0. This equation is true when sin(x) = 0 or cos(x) = 0.
When sin(x) = 0, the solutions are x = 0, pi, and 2pi.
When cos(x) = 0, the solutions are x = pi/2 and 3pi/2.
In summary, the real numbers in the interval (0, 2 pi) where tan(x)sin(2x) = 0 are x = 0, pi/2, pi, 3pi/2, and 2pi.