Let tan(2x) + tan(x) = 0, where 0 ≤ x < 2π. How many possible solutions are there for x?

Use the identity tan(2x) = 2 tan(x)/(1 − tan²(x)), where both tangent expressions must be defined. Let t = tan(x); then the equation becomes t(3 − t²)/(1 − t²) = 0, so t = 0 or t = ±√3. In 0 ≤ x < 2π, tan(x) = 0 gives 2 solutions, and tan(x) = ±√3 gives 4 more, for 6 solutions total. The correct choice is C, 6.

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