What is the general solution for (csc(θ))/(cot(θ))=-2tan(θ) ?

The general solution for (csc(θ))/(cot(θ))=-2tan(θ) is θ=(7pi)/6+2pin,θ=(11pi)/6+2pin

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(csc(θ))/(cot(θ))=-2tan(θ)

\frac{csc ( θ )}{cot ( θ )} = - 2 tan (θ)

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Solution steps

\frac{csc ( θ )}{cot ( θ )} = - 2 tan (θ)

Subtract

- 2 tan (θ)

from both sides

\frac{csc ( θ )}{cot ( θ )} + 2 tan (θ) = 0

Simplify

\frac{csc ( θ )}{cot ( θ )} + 2 tan (θ) : \frac{csc ( θ ) + 2 tan ( θ ) cot ( θ )}{cot ( θ )}

\frac{csc ( θ ) + 2 tan ( θ ) cot ( θ )}{cot ( θ )} = 0

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

csc (θ) + 2 tan (θ) cot (θ) = 0

Rewrite using trig identities

2 + csc (θ) = 0

Move

2

to the right side

csc (θ) = - 2

General solutions for

csc (θ) = - 2

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

2 cos^{2} (θ) + 3 sin (θ) = 0

sin (2 θ) + 3 cos (θ) = 0

2 cot (x) + csc^{2} (x) = 0

2 cos^{2} (θ) + 3 cos (θ) + 4 = 3

2 sin (3 θ) cos (2 θ) - cos (2 θ) = 0

What is the general solution for (csc(θ))/(cot(θ))=-2tan(θ) ?
The general solution for (csc(θ))/(cot(θ))=-2tan(θ) is θ=(7pi)/6+2pin,θ=(11pi)/6+2pin