What is the general solution for 2csc^2(x)-2csc(x)-1=0 ?

The general solution for 2csc^2(x)-2csc(x)-1=0 is x=0.82132…+2pin,x=pi-0.82132…+2pin

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2csc^2(x)-2csc(x)-1=0

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

x = 0.82132 \dots + 2 πn, x = π - 0.82132 \dots + 2 πn

Solution steps

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

Solve by substitution

csc (x) = \frac{1 + 3}{2}, csc (x) = \frac{1 - 3}{2}

csc (x) = \frac{1 + 3}{2} : x = arccsc (\frac{1 + 3}{2}) + 2 πn, x = π - arccsc (\frac{1 + 3}{2}) + 2 πn

csc (x) = \frac{1 - 3}{2} :

No Solution

Combine all the solutions

x = arccsc (\frac{1 + 3}{2}) + 2 πn, x = π - arccsc (\frac{1 + 3}{2}) + 2 πn

Show solutions in decimal form

x = 0.82132 \dots + 2 πn, x = π - 0.82132 \dots + 2 πn

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

cos (x) = \frac{6}{15}

arccos (x) = 90

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

sin (θ) = \frac{7}{12}

What is the general solution for 2csc^2(x)-2csc(x)-1=0 ?
The general solution for 2csc^2(x)-2csc(x)-1=0 is x=0.82132…+2pin,x=pi-0.82132…+2pin