What is the general solution for 3cot^2(x)=7csc(x)-7 ?

The general solution for 3cot^2(x)=7csc(x)-7 is x=0.84806…+2pin,x=pi-0.84806…+2pin,x= pi/2+2pin

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3cot^2(x)=7csc(x)-7

3 cot^{2} (x) = 7 csc (x) - 7

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn, x = \frac{π}{2} + 2 πn

Solution steps

3 cot^{2} (x) = 7 csc (x) - 7

Subtract

7 csc (x) - 7

from both sides

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

Rewrite using trig identities

4 + 3 csc^{2} (x) - 7 csc (x) = 0

Solve by substitution

csc (x) = \frac{4}{3}, csc (x) = 1

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

csc (x) = 1 : x = \frac{π}{2} + 2 πn

Combine all the solutions

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

Show solutions in decimal form

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn, x = \frac{π}{2} + 2 πn

sin (x) = 0.86

2 sin (2 θ) + 3 = 0

sin^{2} (x) + 4 sin (x) + 3 = 0

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

4 cos (2 θ) + 3 cos (θ) + 4 = - 3 cos (θ) - 1

What is the general solution for 3cot^2(x)=7csc(x)-7 ?
The general solution for 3cot^2(x)=7csc(x)-7 is x=0.84806…+2pin,x=pi-0.84806…+2pin,x= pi/2+2pin