What is the general solution for cot^2(x)=-sqrt(3)cot(x) ?

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

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cot^2(x)=-sqrt(3)cot(x)

cot^{2} (x) = - 3 cot (x)

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

Solution steps

cot^{2} (x) = - 3 cot (x)

Solve by substitution

cot (x) = 0, cot (x) = - 3

cot (x) = 0 : x = \frac{π}{2} + πn

cot (x) = - 3 : x = \frac{5 π}{6} + πn

Combine all the solutions

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

tan (θ) = \frac{cos ( θ )}{sin ( θ )}

sin (θ) + 2 sin^{2} (θ) = 0

(cot (x) + 1) (sin (x) + 1) = 0

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

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

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