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

The general solution for -3=cot^2(x)+3csc(x) is x=(3pi)/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin

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-3=cot^2(x)+3csc(x)

- 3 = cot^{2} (x) + 3 csc (x)

x = \frac{3 π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

- 3 = cot^{2} (x) + 3 csc (x)

Switch sides

cot^{2} (x) + 3 csc (x) = - 3

Subtract

- 3

from both sides

cot^{2} (x) + 3 csc (x) + 3 = 0

Rewrite using trig identities

2 + csc^{2} (x) + 3 csc (x) = 0

Solve by substitution

csc (x) = - 1, csc (x) = - 2

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

csc (x) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{3 π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

4 cos^{2} (θ) + 3 cos (θ) = 1

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

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

400 = 812.25 + 400 - 1140 cos (a)

3 cos (θ) = sin (θ)

What is the general solution for -3=cot^2(x)+3csc(x) ?
The general solution for -3=cot^2(x)+3csc(x) is x=(3pi)/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin