SOLUTION: Please help me solve this problem: If {{{ sqrt(3)cot^2 - 4cot + sqrt(3)= 0 }}} Then find cot^2 + tan^2

Question 789080: Please help me solve this problem:
If AMP Parsing Error of [ sqrt(3)cot^2 - 4cot + sqrt(3)= 0 ]: Invalid function '-4\cot+sqrt(3)=0': opening bracket expected at /home/ichudov/project_locations/algebra.com/templates/Algebra/Expression.pm line 70. .
Then find cot^2 + tan^2
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!

Let's make a change of varaible so I do not have to write so many times.

Now I can write the equation as

We have to solve that equation
It may not be obvious, but we can solve that equation by factoring.
The left diside of the equal sign can be factored to get

That leads to 2 solutions
<-->
<-->
Since , the solutions look very symmetrical:
-->
and
-->
One function is and the other is .
For either set of solutions,

PS _ No, I would not have tried to use the quadratic formula to solve
, but I did not think of the factorization right away.
The way I really figured out the solutions was dividing both sides by (same as multiplying times if you like it better that way, and then "completing the square:

Then, taking square roots:
--> -->

EXTRA:
The problem did not ask about the angles, but the angles in the first quadrant that have those tangent and cotangent values are and (or and if you prefer degrees).
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