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

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

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

csc (x) cot (x) = 23

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

Solution steps

csc (x) cot (x) = 23

Subtract

23

from both sides

csc (x) cot (x) - 23 = 0

Express with sin, cos

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

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

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

Rewrite using trig identities

cos (x) - (1 - cos^{2} (x)) \cdot 23 = 0

Solve by substitution

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

cos (x) = \frac{3}{2} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos (x) = - \frac{2 3}{3} :

No Solution

Combine all the solutions

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

sin (θ) cos (θ) - sin (θ) = 0

cos^{2} (θ) + 2 sin (θ) + 1 = 0

arcsec (x) = arcsec (2)

tan^{2} (x) + (3 - 1) tan (x) - 3 = 0

sin (θ) = \frac{2}{7}

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