What is the general solution for sqrt(2)sin(x)-tan(x)=0 ?

The general solution for sqrt(2)sin(x)-tan(x)=0 is x=2pin,x=pi+2pin,x= pi/4+2pin,x=(7pi)/4+2pin

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

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

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Solution steps

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

Express with sin, cos

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

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

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

Factor

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

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

- 1 + 2 cos (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

2 sin^{2} (x) - 13 sin (x) + 6 = 0

2 tan (θ) + 3 = 0

tan (x) = \frac{4}{6}

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

cos^{2} (θ) + 2 cos (θ) = 0

What is the general solution for sqrt(2)sin(x)-tan(x)=0 ?
The general solution for sqrt(2)sin(x)-tan(x)=0 is x=2pin,x=pi+2pin,x= pi/4+2pin,x=(7pi)/4+2pin