What is the general solution for tan(4x)=2tan(2x) ?

The general solution for tan(4x)=2tan(2x) is x=(pin)/2

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tan(4x)=2tan(2x)

tan (4 x) = 2 tan (2 x)

x = \frac{πn}{2}

Solution steps

tan (4 x) = 2 tan (2 x)

Subtract

2 tan (2 x)

from both sides

tan (4 x) - 2 tan (2 x) = 0

Let:

u = 2 x

tan (2 u) - 2 tan (u) = 0

Rewrite using trig identities

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

Solve by substitution

tan (u) = 0

tan (u) = 0 : u = πn

Combine all the solutions

u = πn

Substitute back

u = 2 x

2 x = πn : x = \frac{πn}{2}

x = \frac{πn}{2}

sec (x) - 4 = - sec (x)

5 cos (x) = \frac{1}{cos ( x )}

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

33 + 2 tan (x) = 23 - tan (x)

(csc^{2} (x) - 2) (3 csc^{2} (x) - 4) = 0

What is the general solution for tan(4x)=2tan(2x) ?
The general solution for tan(4x)=2tan(2x) is x=(pin)/2