What is the general solution for cot^2(x)-2csc(x)=-1 ?

The general solution for cot^2(x)-2csc(x)=-1 is x= pi/6+2pin,x=(5pi)/6+2pin

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cot^2(x)-2csc(x)=-1

cot^{2} (x) - 2 csc (x) = - 1

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

Solution steps

cot^{2} (x) - 2 csc (x) = - 1

Subtract

- 1

from both sides

cot^{2} (x) - 2 csc (x) + 1 = 0

Rewrite using trig identities

csc^{2} (x) - 2 csc (x) = 0

Solve by substitution

csc (x) = 2, csc (x) = 0

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

csc (x) = 0 :

No Solution

Combine all the solutions

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

3 csc (x) + csc (x) cot (x) = 0

- 3 cos^{2} (θ) - 10 cos (θ) = - 3 cos (θ) - 6

- 4 sin (x) = - cos^{2} (x) + 4

sin^{2} (x) - 4 sin (x) + 3 = 0

6 sin (x) + csc (x) = 0

What is the general solution for cot^2(x)-2csc(x)=-1 ?
The general solution for cot^2(x)-2csc(x)=-1 is x= pi/6+2pin,x=(5pi)/6+2pin