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

The general solution for tan(2x)-2=3tan(x) is x=0.67351…+pin

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

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

x = 0.67351 \dots + πn

Solution steps

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

Subtract

3 tan (x)

from both sides

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

Rewrite using trig identities

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

Solve by substitution

tan (x) \approx 0.79798 \dots

tan (x) = 0.79798 \dots : x = arctan (0.79798 \dots) + πn

Combine all the solutions

x = arctan (0.79798 \dots) + πn

Show solutions in decimal form

x = 0.67351 \dots + πn

sin (x - \frac{π}{6}) = 0

2 = \frac{1}{cos ^{2} ( x )}

arccot (x - 2) = arccot (x - 1) + arccot (x)

tan (x) - \frac{2}{tan ( x )} = 1

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

What is the general solution for tan(2x)-2=3tan(x) ?
The general solution for tan(2x)-2=3tan(x) is x=0.67351…+pin