What is the general solution for 4csc(x)=sqrt(3)csc(x)cot(x)+3csc(x) ?

The general solution for 4csc(x)=sqrt(3)csc(x)cot(x)+3csc(x) is x= pi/3+pin

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4csc(x)=sqrt(3)csc(x)cot(x)+3csc(x)

4 csc (x) = 3 csc (x) cot (x) + 3 csc (x)

x = \frac{π}{3} + πn

Solution steps

4 csc (x) = 3 csc (x) cot (x) + 3 csc (x)

Subtract

3 csc (x) cot (x) + 3 csc (x)

from both sides

csc (x) - 3 csc (x) cot (x) = 0

Factor

csc (x) - 3 csc (x) cot (x) : csc (x) (1 - 3 cot (x))

csc (x) (1 - 3 cot (x)) = 0

Solving each part separately

csc (x) = 0 or 1 - 3 cot (x) = 0

csc (x) = 0 :

No Solution

1 - 3 cot (x) = 0 : x = \frac{π}{3} + πn

Combine all the solutions

x = \frac{π}{3} + πn

4 cos^{2} (x) + 4 cos (x) = 1, 0^{\circ} \leq x < 36 0^{\circ}

8 cos^{2} (x) = 4

4 cos (3 x) = 0

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

cos (x) = \frac{6}{8}

What is the general solution for 4csc(x)=sqrt(3)csc(x)cot(x)+3csc(x) ?
The general solution for 4csc(x)=sqrt(3)csc(x)cot(x)+3csc(x) is x= pi/3+pin