What is the general solution for 3tan^3(x)-3tan^2(x)-tan(x)+1=0 ?

The general solution for 3tan^3(x)-3tan^2(x)-tan(x)+1=0 is x= pi/4+pin,x=(5pi)/6+pin,x= pi/6+pin

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3tan^3(x)-3tan^2(x)-tan(x)+1=0

3 tan^{3} (x) - 3 tan^{2} (x) - tan (x) + 1 = 0

x = \frac{π}{4} + πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Solution steps

3 tan^{3} (x) - 3 tan^{2} (x) - tan (x) + 1 = 0

Solve by substitution

tan (x) = 1, tan (x) = - \frac{3}{3}, tan (x) = \frac{3}{3}

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = - \frac{3}{3} : x = \frac{5 π}{6} + πn

tan (x) = \frac{3}{3} : x = \frac{π}{6} + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

sin (2 x) = 3 cos (2 x)

cos (2 x) - sin (x) = 1

sec (3 x) = - 1

2 cos^{2} (x) + 1 = 3 cos (x)

cos (x) + 2 = 3 cos (x)

What is the general solution for 3tan^3(x)-3tan^2(x)-tan(x)+1=0 ?
The general solution for 3tan^3(x)-3tan^2(x)-tan(x)+1=0 is x= pi/4+pin,x=(5pi)/6+pin,x= pi/6+pin