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

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

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tan^2(x)=2tan(x)sin(x)

tan^{2} (x) = 2 tan (x) sin (x)

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

Solution steps

tan^{2} (x) = 2 tan (x) sin (x)

Subtract

2 tan (x) sin (x)

from both sides

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

Factor

tan^{2} (x) - 2 tan (x) sin (x) : tan (x) (tan (x) - 2 sin (x))

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

Solving each part separately

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

tan (x) = 0 : x = πn

tan (x) - 2 sin (x) = 0 : x = 2 πn, x = π + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

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

Merge Overlapping Intervals

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

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

sin (u) = - \frac{3}{5}

2 sin^{2} (x) = cos (x) + 1

2 cos^{2} (θ) + cos (θ) = 0

9 sec (x) + 9 tan (x) = 9

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