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

The general solution for tan^2(x)sin(x)-(sin(x))/3 =0 is x=2pin,x=pi+2pin,x=(5pi)/6+pin,x= pi/6+pin

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tan^2(x)sin(x)-(sin(x))/3 =0

tan^{2} (x) sin (x) - \frac{sin ( x )}{3} = 0

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

Solution steps

tan^{2} (x) sin (x) - \frac{sin ( x )}{3} = 0

Simplify

tan^{2} (x) sin (x) - \frac{sin ( x )}{3} : \frac{3 tan ^{2} ( x ) sin ( x ) - sin ( x )}{3}

\frac{3 tan ^{2} ( x ) sin ( x ) - sin ( x )}{3} = 0

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

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

Factor

3 tan^{2} (x) sin (x) - sin (x) : sin (x) (3 tan (x) + 1) (3 tan (x) - 1)

sin (x) (3 tan (x) + 1) (3 tan (x) - 1) = 0

Solving each part separately

sin (x) = 0 or 3 tan (x) + 1 = 0 or 3 tan (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

3 tan (x) + 1 = 0 : x = \frac{5 π}{6} + πn

3 tan (x) - 1 = 0 : x = \frac{π}{6} + πn

Combine all the solutions

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

cot^{2} (a) = cos^{2} (a) (cot (a) cos (a))^{2}

1 + cot^{2} (x) = 8 sin (x)

4 sin^{2} (x) = 8 sin^{2} (\frac{x}{2})

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

cos (\frac{π}{2} - x) = 0

What is the general solution for tan^2(x)sin(x)-(sin(x))/3 =0 ?
The general solution for tan^2(x)sin(x)-(sin(x))/3 =0 is x=2pin,x=pi+2pin,x=(5pi)/6+pin,x= pi/6+pin