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There may be other ways to solve this but what came to me first was to look at this as a polynomial in tan(x). It might help you to see this if I use a temporary variable:
Let q = tan(x)
Substituting q for tan(x) we get:
To solve this we factor it. There is no GCF and there are too many terms for pattern factoring or trinomial factoring. Factoring by grouping will work but it is difficult. And factoring by trial and error of the possible rational roots will work, too. I'll do it both ways in hopes you might become better at factoring.
Factoring by grouping.
Rearrange the terms:
Group:
Factor the first group as a sum of cubes. Factor out a GCF of -3q from the second group:
Factor out the common factor of (q+1):
The second factor simplifies:
Factoring by trial and error of the possible rational roots.
The only possible rational roots are 1 and -1. 1 is not an actual factor. But -1 is. We can see this by using synthetic division to divide our polynomial by (q-(-1)) or just (q+1):
The zero in the lower right is the remainder. Since it is zero, (q+1) divides equally. (IOW, it is a factor.) The rest of the bottom row tells us the other factor. The "1 -4 1" translates into
So either way we factor our polynomial we end up with
The quadratic factor will not factor further. From the Zero Product Property:
q+1 = 0 or
The first equation solves to q = -1. For the second equation we use the Quadratic formula:
Now's the time we replace the q's with tan(x):
tan(x) = -1
or
or
In the equation tan(x) = -1, we should recognize -1 as a special angle value for tan. We should know that the reference angle would be and that since tan is negative in the 2nd and 4th quadrants...
tan(x) = -1
leads us to: (for the 2nd quadrant)
or (for the 4th quadrant)
These simplify to: (for the 2nd quadrant)
or (for the 4th quadrant)
In the other equations:
or
we should recognize that these are not special angle values for tan. (If the 2's were not there then we would have special angle values!) So we will need to use our calculators. First we will get decimals for these tan values:
or
Let's deal with first. Using the inverse tan button on our calculator: (If you don't get this, then maybe your calculator is not set to radian mode. Try multiplying your answer by and see if you get the decimal above.) So we have a reference angle of 1.30899694. Since tan is positive in the 1st and 3rd quadrants we get: (for the 1st quadrant)
or (for the 3rd quadrant.
Using the same steps on
we should get a reference angle of 0.26179939 and the equations: (for the 1st quadrant)
or (for the 3rd quadrant.
Putting all our solutions all together we have:
or
or
or
or
or
All of these together are the general solution to:
(