Question

Find the general solution of the following equation: 

tan³θ = 3 tanθ.

Answer 1

tan³θ = 3tanθ
∴ tan³θ - 3tanθ = 0
∴ tanθ (tan²θ - 3) = 0
∴ either tanθ = 0 or tan²θ - 3 = 0
∴ either tanθ = 0 or tan²θ = 3
∴ either tanθ = 0 or tan²θ = (√3)²
∴ either tan θ = 0 or tan²θ = `(tan  pi/3)^2    ...[∵ tan  pi/(3) = sqrt(3)]`
∴ either tanθ = 0 or tan²θ = `tan^2  pi/(3)`

The  general solution of 
tanθ = 0 is θ = nπ, n ∈ Z and 
tan²θ = tan²α is θ = nπ ± α, n ∈ Z.

∴ the required general solution is given by
θ = nπ, n ∈ Z or θ = `nπ ± pi/(3)`, n ∈ Z.