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

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

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

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

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

Solution steps

tan (x) - 2 sin (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

4 sin (2 t) = 4 cos (t)

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

1 + 2 sin^{2} (t) = - 5 cos (t)

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

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

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