Verified answer

csc²Θ - 6 = cscΘ

let cscΘ = x

x² - 6 = x

x² - x - 6 = 0

factoring:

(x - 3)(x + 2) = 0

so

x = 3

or

x = -2

plugging back in cscΘ = x

cscΘ = 3

or

cscΘ = -2

cscΘ = 1/(sinΘ)

1/(sinΘ) = 3

or

1/(sinΘ) = -2

sinΘ = 1/3

or

sinΘ = -1/2

Θ = arcsin (1/3) = 19.5 degrees

or

Θ = arcsin (-1/2) = 330 degrees

or 1/sin^2Θ - 6 = 1/Sin Θ

or1-6Sin^2 Θ - SinΘ =0

or 6Sin^2 Θ +SinΘ -1 =0

or 6 Sin^2Θ +3SinΘ -2sinΘ -1=0

or 3SinΘ(2SinΘ + 1) - 1(2SinΘ + 1)=0

or (3SinΘ -1)(2SinΘ + 1) = 0

so Θ=Sin-1(1/3) or Θ= Sin-1(-1/2)

so Θ = 19.3 + 2n*pi deg or Θ = 330 + 2n*pi deg

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