Thank you for the opportunity to help you with your question!
1. Think of the equation as a quadratic, substitute x instead of tan(x).
6 x^2 - 17 x + 7 = 0. (given it is a quadratic we can factor it)
(3 x - 7) (2x -1) = 0 (but since the original uses tan instead of x, we substitute tan(X) back in)
(3 tan (x) - 7) ( 2 tan(x) -1) = 0 . (now we solve for x for each equation)
3 tan (x) = 7 which is tan(x) = 7/3 and 2 tan(x) = 1 which is tan(x) = 1/2
Using a calculator we take the arc tangent of the values (in radian mode) and our answer is
1.166 radians and 0.46364 radians .
BUT! Since tangent is positive in both 1st and 3rd quadrant, there are 2 answers for each. To find the answer, we have to add Pi radians to both answers, or 3.1415.
So the 4 answers are: 1.166 radians, 4.3075 radians, 0.4364 radians, and 3.578 radians
2. sin (x) = -2/3
using a graphical approach, we know there are 2 answer, one in the 3rd and one in the 4th quadrant
using arc sin(X), we get the value to be -0.7297 radians. Since its from 0 to 2pi, we add the value (since it is negative) to 2 pi.
6.2832 radians + -0.7297 radians, gives us: 5.553 radians
Since the other answer is in the 3rd quadrant, what we do is add the 0.7297 value to 1pi to get the 3rd quadrant answer. So:
3.1415 radians + 0.7297 radians gives us: 3.8713 radians
3. solve so tan(2x) is by itself.
tan 2x = 1
tan (x) = 1 at pi/4 and 5pi/4. so then we have to find half of those respective values
meaning your answers are pi/8 and 5pi/8
4. 2 answers
cot (u) = -1
cot(u) = -1 when sin and cos are the same, except one is positive and the other negative, meaning 2nd and 4th quadrant. That is true at 3pi/4 and 7pi/4.
csc u = 1. csc is the same thing as 1/sin. So for that to be true, sine must also be 1, because 1/1 = 1.
That is only true at 1 point, (pi/2)
Please let me know if you need any clarification. I'm always happy to answer your questions.