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

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

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tan(2x)-3tan(x)=0

tan (2 x) - 3 tan (x) = 0

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

Solution steps

tan (2 x) - 3 tan (x) = 0

Rewrite using trig identities

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

Solve by substitution

tan (x) = 0, tan (x) = - \frac{3}{3}, tan (x) = \frac{3}{3}

tan (x) = 0 : x = 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 = πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

4 sin (x) - 3 cos (x) = 0

25 cos^{2} (3 x) + 10 cos (3 x) = 15

5 cos^{2} (x) = 6 sin (x)

sec^{2} (x) = 2 tan (x) + 4

- 4 cos (θ) = - sin^{2} (θ) + 1

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