What is the general solution for 2sin(2x)-3-6/(sin(2x)-1)=0 ?

The general solution for 2sin(2x)-3-6/(sin(2x)-1)=0 is x=(7pi)/(12)+pin,x=(11pi)/(12)+pin

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2sin(2x)-3-6/(sin(2x)-1)=0

2 sin (2 x) - 3 - \frac{6}{sin ( 2 x ) - 1} = 0

x = \frac{7 π}{12} + πn, x = \frac{11 π}{12} + πn

Solution steps

2 sin (2 x) - 3 - \frac{6}{sin ( 2 x ) - 1} = 0

Solve by substitution

sin (2 x) = 3, sin (2 x) = - \frac{1}{2}

sin (2 x) = 3 :

No Solution

sin (2 x) = - \frac{1}{2} : x = \frac{7 π}{12} + πn, x = \frac{11 π}{12} + πn

Combine all the solutions

x = \frac{7 π}{12} + πn, x = \frac{11 π}{12} + πn

3 sin^{2} (x) + 11 sin (x) + 10 = 0

cos (x) (cos (x) + 5) = 2 sin^{2} (x)

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

cos^{2} (x) - sin (x) = - 1

4 sin (x) - 3 sin^{2} (x) + cos^{2} (x) - 2 = 0

What is the general solution for 2sin(2x)-3-6/(sin(2x)-1)=0 ?
The general solution for 2sin(2x)-3-6/(sin(2x)-1)=0 is x=(7pi)/(12)+pin,x=(11pi)/(12)+pin