What is the general solution for 2sin(x)csc(x)-csc(x)=4sin(x)-2 ?

The general solution for 2sin(x)csc(x)-csc(x)=4sin(x)-2 is x= pi/6+2pin,x=(5pi)/6+2pin

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2sin(x)csc(x)-csc(x)=4sin(x)-2

2 sin (x) csc (x) - csc (x) = 4 sin (x) - 2

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

2 sin (x) csc (x) - csc (x) = 4 sin (x) - 2

Subtract

4 sin (x) - 2

from both sides

2 sin (x) csc (x) - csc (x) - 4 sin (x) + 2 = 0

Factor

2 sin (x) csc (x) - csc (x) - 4 sin (x) + 2 : (2 sin (x) - 1) (csc (x) - 2)

(2 sin (x) - 1) (csc (x) - 2) = 0

Solving each part separately

2 sin (x) - 1 = 0 or csc (x) - 2 = 0

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) - 2 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 cos^{2} (x) + 5 cos (x) + 2 = 0

sin (θ) = - 0.66

4 sec^{2} (θ) tan (θ) = 8 tan (θ)

5 sec (x) csc (x) = 10 csc (x)

cos (x) = 1 - sin (x)

What is the general solution for 2sin(x)csc(x)-csc(x)=4sin(x)-2 ?
The general solution for 2sin(x)csc(x)-csc(x)=4sin(x)-2 is x= pi/6+2pin,x=(5pi)/6+2pin