What is the general solution for csc^2(x)+6csc(x)-7=0 ?

The general solution for csc^2(x)+6csc(x)-7=0 is x= pi/2+2pin,x=-0.14334…+2pin,x=pi+0.14334…+2pin

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csc^2(x)+6csc(x)-7=0

csc^{2} (x) + 6 csc (x) - 7 = 0

x = \frac{π}{2} + 2 πn, x = - 0.14334 \dots + 2 πn, x = π + 0.14334 \dots + 2 πn

Solution steps

csc^{2} (x) + 6 csc (x) - 7 = 0

Solve by substitution

csc (x) = 1, csc (x) = - 7

csc (x) = 1 : x = \frac{π}{2} + 2 πn

csc (x) = - 7 : x = arccsc (- 7) + 2 πn, x = π + arccsc (7) + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = arccsc (- 7) + 2 πn, x = π + arccsc (7) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = - 0.14334 \dots + 2 πn, x = π + 0.14334 \dots + 2 πn

\frac{sin ( 2 x ) - 1}{3} = - \frac{1}{6}

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

4 cos (x) + 2 = 3

sin (4 θ) = 4 sin (2 θ) cos (2 θ)

cos (x) = 6

What is the general solution for csc^2(x)+6csc(x)-7=0 ?
The general solution for csc^2(x)+6csc(x)-7=0 is x= pi/2+2pin,x=-0.14334…+2pin,x=pi+0.14334…+2pin