What is the general solution for 3csc^2(x)+1=5 ?

The general solution for 3csc^2(x)+1=5 is x= pi/3+2pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin,x=(5pi)/3+2pin

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3csc^2(x)+1=5

3 csc^{2} (x) + 1 = 5

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Solution steps

3 csc^{2} (x) + 1 = 5

Solve by substitution

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

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

csc (x) = - \frac{2 3}{3} : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

6 cos^{2} (x) - 12 cos (x) + 6 = 0

csc (x) = sin (x) tan (x) + cos (x)

6 sin^{2} (θ) - 10 sin (θ) - 8 = - 3 sin (θ) - 3

sin (4 θ) - sin (6 θ) = 0

sin (x) = \frac{3}{10}

What is the general solution for 3csc^2(x)+1=5 ?
The general solution for 3csc^2(x)+1=5 is x= pi/3+2pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin,x=(5pi)/3+2pin