What is the general solution for 3cos(x)tan^2(x)=cos(x) ?

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

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3cos(x)tan^2(x)=cos(x)

3 cos (x) tan^{2} (x) = cos (x)

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

Solution steps

3 cos (x) tan^{2} (x) = cos (x)

Subtract

cos (x)

from both sides

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

Factor

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

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

Solving each part separately

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

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 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 = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn

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

cos (x) = cos (3 x)

sin (θ) = tan (θ)

sin (x) = \frac{1}{8}

2 cos (2 θ) = 1

8 cos (θ) - 43 = 0

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