Question

Solve the equation on the interval [0, 2π). tan^2x sin x = tan^2x

Answer 1

To solve the equation tan^2x sin x = tan^2x on the interval [0, 2π), divide both sides by tan^2x. The solution is x = π/2.

Step-by-step explanation:

To solve the equation tan^2x sin x = tan^2x on the interval [0, 2π), we will first simplify the equation by dividing both sides by tan^2x. This gives us the equation sin x = 1.

Next, we need to find the values of x in the given interval that make the equation true. Since sin x = 1 only when x = π/2, we can conclude that the solution to the equation on the interval [0, 2π) is x = π/2.

Answer 2

Start by getting everything on the same side of the equals sign and then set it equal to 0.
`tan^2xsinx-tan^2x=0`. Factor out the common tan^2x like this:
`tan^2x(sinx-1)=0`. Now we have 2 separate equations to solve:
`tan^2x=0` and sinx = 0. Now we have to figure out where tan^2 is 0 between 0 and 2pi. If we include 2pi, the solutions for that equation are
`x = 0, \pi , 2 \pi`. You can test those out on your calculator just to be sure. There's only one value of x for the next equation. The only place between 0 and 2pi where the sin x = 1 is at x =
`( \pi )/(2)`. And there you go!