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

The general solution for sin(2x)=2tan(2x) is x=pin,x= pi/2+pin

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sin(2x)=2tan(2x)

sin (2 x) = 2 tan (2 x)

x = πn, x = \frac{π}{2} + πn

Solution steps

sin (2 x) = 2 tan (2 x)

Subtract

2 tan (2 x)

from both sides

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

Express with sin, cos

\frac{- 2 sin ( 2 x ) + cos ( 2 x ) sin ( 2 x )}{cos ( 2 x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 2 sin (2 x) + cos (2 x) sin (2 x) = 0

Factor

- 2 sin (2 x) + cos (2 x) sin (2 x) : sin (2 x) (cos (2 x) - 2)

sin (2 x) (cos (2 x) - 2) = 0

Solving each part separately

sin (2 x) = 0 or cos (2 x) - 2 = 0

sin (2 x) = 0 : x = πn, x = \frac{π}{2} + πn

cos (2 x) - 2 = 0 :

No Solution

Combine all the solutions

x = πn, x = \frac{π}{2} + πn

- 2 cos (2 θ) + 12 = 4 sin (θ) + 9

sec^{2} (x) + sec (x) - 2 = 0

(cot (x) - 1) (2 sin (x) + 1) = 0

y, x = tan (y)

tan (x) = - \frac{12}{5}

What is the general solution for sin(2x)=2tan(2x) ?
The general solution for sin(2x)=2tan(2x) is x=pin,x= pi/2+pin