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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(csc(x))/(cot(x))=csc(x)sec(x)

Solution

\frac{csc ( x )}{cot ( x )} = csc (x) sec (x)

Solution

No Solution for x \in R

Solution steps

\frac{csc ( x )}{cot ( x )} = csc (x) sec (x)

Subtract

csc (x) sec (x)

from both sides

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

Simplify

\frac{csc ( x )}{cot ( x )} - csc (x) sec (x) : \frac{csc ( x ) - csc ( x ) sec ( x ) cot ( x )}{cot ( x )}

\frac{csc ( x )}{cot ( x )} - csc (x) sec (x)

Convert element to fraction:

csc (x) sec (x) = \frac{csc ( x ) sec ( x ) cot ( x )}{cot ( x )}

= \frac{csc ( x )}{cot ( x )} - \frac{csc ( x ) sec ( x ) cot ( x )}{cot ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{csc ( x ) - csc ( x ) sec ( x ) cot ( x )}{cot ( x )}

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

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

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

Factor

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

csc (x) - csc (x) sec (x) cot (x)

Factor out common term

csc (x)

= csc (x) (1 - sec (x) cot (x))

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

Solving each part separately

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

csc (x) = 0 :

No Solution

csc (x) = 0

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

1 - sec (x) cot (x) = 0 :

No Solution

1 - sec (x) cot (x) = 0

Express with sin, cos

1 - cot (x) sec (x)

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

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

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

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

Simplify

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

1 - \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} = \frac{1}{sin ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{cos ( x ) \cdot 1}{sin ( x ) cos ( x )}

Cancel the common factor:

cos (x)

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

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

Convert element to fraction:

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

= \frac{1 \cdot sin ( x )}{sin ( x )} - \frac{1}{sin ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot sin ( x ) - 1}{sin ( x )}

Multiply:

1 \cdot sin (x) = sin (x)

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

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

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

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

- 1 + sin (x) = 0

Move

1

to the right side

- 1 + sin (x) = 0

Add

1

to both sides

- 1 + sin (x) + 1 = 0 + 1

Simplify

sin (x) = 1

sin (x) = 1

General solutions for

sin (x) = 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn

No Solution

Combine all the solutions

No Solution for x \in R

Graph

Popular Examples

sin(θ)= 1/(sqrt(3))

sin (θ) = \frac{1}{3}

cos(θ)=0.9

cos (θ) = 0.9

tan^2(x)=2tan(x)

tan^{2} (x) = 2 tan (x)

4-4sin(θ)=4

4 - 4 sin (θ) = 4

4cos(x)sin(x)=-5sin(x)

4 cos (x) sin (x) = - 5 sin (x)

Frequently Asked Questions (FAQ)

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