What is the general solution for (cos(x)cot(x))/(1-sin(x))=3 ?

The general solution for (cos(x)cot(x))/(1-sin(x))=3 is x= pi/6+2pin,x=(5pi)/6+2pin

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(cos(x)cot(x))/(1-sin(x))=3

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} = 3

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} = 3

Subtract

3

from both sides

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

Simplify

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} - 3 : \frac{cos ( x ) cot ( x ) - 3 ( 1 - sin ( x ) )}{1 - sin ( x )}

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

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

cos (x) cot (x) - 3 (1 - sin (x)) = 0

Rewrite using trig identities

- 3 + \frac{1 + 2 sin ^{2} ( x )}{sin ( x )} = 0

Solve by substitution

sin (x) = 1, sin (x) = \frac{1}{2}

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (2 x) sin (x) - cos (x) = 0

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

tan (2 θ) - tan (θ) = 0

cos (t) = - \frac{1}{2}

cos (2 θ) + cos (θ) = 0

What is the general solution for (cos(x)cot(x))/(1-sin(x))=3 ?
The general solution for (cos(x)cot(x))/(1-sin(x))=3 is x= pi/6+2pin,x=(5pi)/6+2pin