Question

How do you solve `sin3x=cos3x*tan3x` for x in the interval [0,2pi)?

Answer 1

The given equation is true for all values in the range `[0,2pi)`

Explanation:

You could (temporarily) replace `3x` with `theta`
so the given equation would look like:
`color(white)("XXX")sin(theta)=cos(theta)*tan(theta)`

But we know that `tan(theta)=(sin(theta))/(cos(theta))`

So we have that the given equation is equivalent to
`color(white)("XXX")sin(theta)=(cancel(cos(theta)))/color(white)("X")*(sin(theta))/(cancel(cos(theta)))`

Which is true for all values of `theta` and therefore true for all values of `x`.