What is the general solution for csc^2(θ)+4csc(θ)-5=0 ?

The general solution for csc^2(θ)+4csc(θ)-5=0 is θ= pi/2+2pin,θ=-0.20135…+2pin,θ=pi+0.20135…+2pin

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csc^2(θ)+4csc(θ)-5=0

csc^{2} (θ) + 4 csc (θ) - 5 = 0

θ = \frac{π}{2} + 2 πn, θ = - 0.20135 \dots + 2 πn, θ = π + 0.20135 \dots + 2 πn

Solution steps

csc^{2} (θ) + 4 csc (θ) - 5 = 0

Solve by substitution

csc (θ) = 1, csc (θ) = - 5

csc (θ) = 1 : θ = \frac{π}{2} + 2 πn

csc (θ) = - 5 : θ = arccsc (- 5) + 2 πn, θ = π + arccsc (5) + 2 πn

Combine all the solutions

θ = \frac{π}{2} + 2 πn, θ = arccsc (- 5) + 2 πn, θ = π + arccsc (5) + 2 πn

Show solutions in decimal form

θ = \frac{π}{2} + 2 πn, θ = - 0.20135 \dots + 2 πn, θ = π + 0.20135 \dots + 2 πn

cos^{2} (x) - cos (x) = 0, 0 \leq x \leq 2 π

\frac{sin ( 2 x )}{cos ( x )} = 2, 0 \leq x \leq π

sec (θ) = - 4

sin (x^{2} + 1) = 0

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

What is the general solution for csc^2(θ)+4csc(θ)-5=0 ?
The general solution for csc^2(θ)+4csc(θ)-5=0 is θ= pi/2+2pin,θ=-0.20135…+2pin,θ=pi+0.20135…+2pin