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

The general solution for tan(x/2)*tan(x)=-3 is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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

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

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Solution steps

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

Subtract

- 3

from both sides

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

Let:

u = \frac{x}{2}

tan (u) tan (2 u) + 3 = 0

Rewrite using trig identities

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

Solve by substitution

tan (u) = 3, tan (u) = - 3

tan (u) = 3 : u = \frac{π}{3} + πn

tan (u) = - 3 : u = \frac{2 π}{3} + πn

Combine all the solutions

u = \frac{π}{3} + πn, u = \frac{2 π}{3} + πn

Substitute back

u = \frac{x}{2}

\frac{x}{2} = \frac{π}{3} + πn : x = \frac{2 π}{3} + 2 πn

\frac{x}{2} = \frac{2 π}{3} + πn : x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x, sin (x) cos (x) = 1

\frac{3 cos ( 2 x ) + 7 sin ( x ) - 5}{9 cos ^{2} ( x ) - 5} = 0

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

sin^{2} (x) + cos^{2} (\frac{17}{9}) = 1

8 sin^{2} (x) - 10 sin (x) + 3 = 0

What is the general solution for tan(x/2)*tan(x)=-3 ?
The general solution for tan(x/2)*tan(x)=-3 is x=(2pi)/3+2pin,x=(4pi)/3+2pin