What is the general solution for cot(θ)+csc(θ)=-sqrt(3) ?

The general solution for cot(θ)+csc(θ)=-sqrt(3) is θ=(5pi)/3+2pin

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cot(θ)+csc(θ)=-sqrt(3)

cot (θ) + csc (θ) = - 3

θ = \frac{5 π}{3} + 2 πn

Solution steps

cot (θ) + csc (θ) = - 3

Subtract

- 3

from both sides

cot (θ) + csc (θ) + 3 = 0

Express with sin, cos

\frac{cos ( θ )}{sin ( θ )} + \frac{1}{sin ( θ )} + 3 = 0

Simplify

\frac{cos ( θ )}{sin ( θ )} + \frac{1}{sin ( θ )} + 3 : \frac{cos ( θ ) + 1 + 3 sin ( θ )}{sin ( θ )}

\frac{cos ( θ ) + 1 + 3 sin ( θ )}{sin ( θ )} = 0

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

cos (θ) + 1 + 3 sin (θ) = 0

Subtract

3 sin (θ)

from both sides

cos (θ) + 1 = - 3 sin (θ)

Square both sides

(cos (θ) + 1)^{2} = (- 3 sin (θ))^{2}

Subtract

(- 3 sin (θ))^{2}

from both sides

(cos (θ) + 1)^{2} - 3 sin^{2} (θ) = 0

Rewrite using trig identities

- 2 + 2 cos (θ) + 4 cos^{2} (θ) = 0

Solve by substitution

cos (θ) = \frac{1}{2}, cos (θ) = - 1

cos (θ) = \frac{1}{2} : θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

cos (θ) = - 1 : θ = π + 2 πn

Combine all the solutions

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

Verify solutions by plugging them into the original equation

θ = \frac{5 π}{3} + 2 πn

3 tan^{2} (θ) + 1 = \frac{2}{tan ^{2} ( θ )}

4 tan (x) = 2 sec^{2} (x)

sin (x) + cos (x) + cot (x) = csc (x)

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

cos (x) = \frac{3}{13}

What is the general solution for cot(θ)+csc(θ)=-sqrt(3) ?
The general solution for cot(θ)+csc(θ)=-sqrt(3) is θ=(5pi)/3+2pin