(2tan(3x))/(1-tan^2(3x))=1

Solution

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

x = - \frac{1.17809 \dots}{3} + \frac{πn}{3}, x = \frac{0.39269 \dots}{3} + \frac{πn}{3}

Solution steps

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

Solve by substitution

tan (3 x) = - 1 - 2, tan (3 x) = 2 - 1

tan (3 x) = - 1 - 2 : x = - \frac{arctan ( 1 + 2 )}{3} + \frac{πn}{3}

tan (3 x) = 2 - 1 : x = \frac{arctan ( 2 - 1 )}{3} + \frac{πn}{3}

Combine all the solutions

x = - \frac{arctan ( 1 + 2 )}{3} + \frac{πn}{3}, x = \frac{arctan ( 2 - 1 )}{3} + \frac{πn}{3}

Show solutions in decimal form

x = - \frac{1.17809 \dots}{3} + \frac{πn}{3}, x = \frac{0.39269 \dots}{3} + \frac{πn}{3}

sin (x) = \frac{3}{3}

cos (2 θ) = - \frac{1}{2}, 0 \leq θ < 2 π

tan (x) tan (2 x) = 1

2 sin^{2} (x) + 7 sin (x) = 4

22 cos (θ) + 4 = 6