What is the general solution for 2sin(x)tan(x)-tan(x)=1-2sin(x) ?

The general solution for 2sin(x)tan(x)-tan(x)=1-2sin(x) is x= pi/6+2pin,x=(5pi)/6+2pin,x=(3pi)/4+pin

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2sin(x)tan(x)-tan(x)=1-2sin(x)

2 sin (x) tan (x) - tan (x) = 1 - 2 sin (x)

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{4} + πn

Solution steps

2 sin (x) tan (x) - tan (x) = 1 - 2 sin (x)

Subtract

1 - 2 sin (x)

from both sides

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

Factor

2 sin (x) tan (x) - tan (x) - 1 + 2 sin (x) : (2 sin (x) - 1) (tan (x) + 1)

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

Solving each part separately

2 sin (x) - 1 = 0 or tan (x) + 1 = 0

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

tan (x) + 1 = 0 : x = \frac{3 π}{4} + πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{4} + πn

tan (θ) = \frac{2 3}{3} sin (θ)

tan (4 x) = - 1

3 (1 - cos (3 x)) = sin^{2} (3 x)

sin (2 x) + 3 sin (x) = 0

cos (2 θ) + 10 sin^{2} (θ) = 5

What is the general solution for 2sin(x)tan(x)-tan(x)=1-2sin(x) ?
The general solution for 2sin(x)tan(x)-tan(x)=1-2sin(x) is x= pi/6+2pin,x=(5pi)/6+2pin,x=(3pi)/4+pin