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

The general solution for (cos(x)cot(x))/(1-sin(x))=csc(x) is No Solution for x\in\mathbb{R}

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

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

No Solution for x \in R

Solution steps

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

Subtract

csc (x)

from both sides

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

Simplify

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

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

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

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

Express with sin, cos

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

Simplify

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

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

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

cos^{2} (x) - 1 + sin (x) = 0

Subtract

sin (x)

from both sides

cos^{2} (x) - 1 = - sin (x)

Square both sides

(cos^{2} (x) - 1)^{2} = (- sin (x))^{2}

Subtract

(- sin (x))^{2}

from both sides

(cos^{2} (x) - 1)^{2} - sin^{2} (x) = 0

Factor

(cos^{2} (x) - 1)^{2} - sin^{2} (x) : (cos^{2} (x) - 1 + sin (x)) (cos^{2} (x) - 1 - sin (x))

(cos^{2} (x) - 1 + sin (x)) (cos^{2} (x) - 1 - sin (x)) = 0

Solving each part separately

cos^{2} (x) - 1 + sin (x) = 0 or cos^{2} (x) - 1 - sin (x) = 0

cos^{2} (x) - 1 + sin (x) = 0 : x = 2 πn, x = π + 2 πn, x = \frac{π}{2} + 2 πn

cos^{2} (x) - 1 - sin (x) = 0 : x = \frac{3 π}{2} + 2 πn, x = 2 πn, x = π + 2 πn

Combine all the solutions

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

Verify solutions by plugging them into the original equation

x = 2 πn, x = π + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

No Solution for x \in R

2 cos^{2} (x) - 9 sin (x) - 3 = 0

3 + \frac{1}{5} \cdot cos (\frac{θ}{3} + \frac{7 π}{4}) = 3

cosh (2 x) = 2 cosh (x) - 1

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

4 cos (3 x) = 2

What is the general solution for (cos(x)cot(x))/(1-sin(x))=csc(x) ?
The general solution for (cos(x)cot(x))/(1-sin(x))=csc(x) is No Solution for x\in\mathbb{R}