What is the general solution for csc(x)cot(x)= 2/3 ?

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

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csc(x)cot(x)= 2/3

csc (x) cot (x) = \frac{2}{3}

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

Solution steps

csc (x) cot (x) = \frac{2}{3}

Subtract

\frac{2}{3}

from both sides

csc (x) cot (x) - \frac{2}{3} = 0

Simplify

csc (x) cot (x) - \frac{2}{3} : \frac{3 csc ( x ) cot ( x ) - 2}{3}

\frac{3 csc ( x ) cot ( x ) - 2}{3} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 csc (x) cot (x) - 2 = 0

Express with sin, cos

3 \cdot \frac{1}{sin ( x )} \cdot \frac{cos ( x )}{sin ( x )} - 2 = 0

Simplify

3 \cdot \frac{1}{sin ( x )} \cdot \frac{cos ( x )}{sin ( x )} - 2 : \frac{3 cos ( x ) - 2 sin ^{2} ( x )}{sin ^{2} ( x )}

\frac{3 cos ( x ) - 2 sin ^{2} ( x )}{sin ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Add

2 sin^{2} (x)

to both sides

3 cos (x) = 2 sin^{2} (x)

Square both sides

(3 cos (x))^{2} = (2 sin^{2} (x))^{2}

Subtract

(2 sin^{2} (x))^{2}

from both sides

9 cos^{2} (x) - 4 sin^{4} (x) = 0

Factor

9 cos^{2} (x) - 4 sin^{4} (x) : (3 cos (x) + 2 sin^{2} (x)) (3 cos (x) - 2 sin^{2} (x))

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

Solving each part separately

3 cos (x) + 2 sin^{2} (x) = 0 or 3 cos (x) - 2 sin^{2} (x) = 0

3 cos (x) + 2 sin^{2} (x) = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

3 cos (x) - 2 sin^{2} (x) = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

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

Verify solutions by plugging them into the original equation

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

2 + 2 cos (x) = 0

4 tan^{2} (x) - 8 tan (x) + 3 = 0

sin (t) - sin (2 t) = 0

sin (\frac{3 x}{2}) = 0

tan (x) = 5 sin (x)

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