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

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

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cot(u)-csc(u)=(sin(u))/(1+cos(u))

Solution

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

Solution

No Solution for u \in R

Solution steps

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

Subtract

\frac{sin ( u )}{1 + cos ( u )}

from both sides

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

Simplify

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

cot (u) - csc (u) - \frac{sin ( u )}{1 + cos ( u )}

Convert element to fraction:

cot (u) = \frac{cot ( u ) ( 1 + cos ( u ) )}{1 + cos ( u )}, csc (u) = \frac{csc ( u ) ( 1 + cos ( u ) )}{1 + cos ( u )}

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

Since the denominators are equal, combine the fractions:

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

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

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

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

cot (u) (1 + cos (u)) - csc (u) (1 + cos (u)) - sin (u) = 0

Express with sin, cos

- sin (u) + (1 + cos (u)) cot (u) - (1 + cos (u)) csc (u)

Use the basic trigonometric identity:

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

= - sin (u) + (1 + cos (u)) \frac{cos ( u )}{sin ( u )} - (1 + cos (u)) csc (u)

Use the basic trigonometric identity:

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

= - sin (u) + (1 + cos (u)) \frac{cos ( u )}{sin ( u )} - (1 + cos (u)) \frac{1}{sin ( u )}

Simplify

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

- sin (u) + (1 + cos (u)) \frac{cos ( u )}{sin ( u )} - (1 + cos (u)) \frac{1}{sin ( u )}

(1 + cos (u)) \frac{cos ( u )}{sin ( u )} = \frac{cos ( u ) ( 1 + cos ( u ) )}{sin ( u )}

(1 + cos (u)) \frac{cos ( u )}{sin ( u )}

Multiply fractions:

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

= \frac{cos ( u ) ( 1 + cos ( u ) )}{sin ( u )}

(1 + cos (u)) \frac{1}{sin ( u )} = \frac{1 + cos ( u )}{sin ( u )}

(1 + cos (u)) \frac{1}{sin ( u )}

Multiply fractions:

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

= \frac{1 \cdot ( 1 + cos ( u ) )}{sin ( u )}

1 \cdot (1 + cos (u)) = 1 + cos (u)

1 \cdot (1 + cos (u))

Multiply:

1 \cdot (1 + cos (u)) = (1 + cos (u))

= (1 + cos (u))

Remove parentheses:

(a) = a

= 1 + cos (u)

= \frac{1 + cos ( u )}{sin ( u )}

= - sin (u) + \frac{cos ( u ) ( cos ( u ) + 1 )}{sin ( u )} - \frac{cos ( u ) + 1}{sin ( u )}

Combine the fractions

\frac{cos ( u ) ( cos ( u ) + 1 )}{sin ( u )} - \frac{cos ( u ) + 1}{sin ( u )} : \frac{cos ( u ) ( 1 + cos ( u ) ) - ( 1 + cos ( u ) )}{sin ( u )}

Apply rule

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

= \frac{cos ( u ) ( cos ( u ) + 1 ) - ( cos ( u ) + 1 )}{sin ( u )}

= - sin (u) + \frac{cos ( u ) ( cos ( u ) + 1 ) - ( cos ( u ) + 1 )}{sin ( u )}

Expand

cos (u) (1 + cos (u)) - (1 + cos (u)) : cos^{2} (u) - 1

cos (u) (1 + cos (u)) - (1 + cos (u))

Expand

cos (u) (1 + cos (u)) : cos (u) + cos^{2} (u)

cos (u) (1 + cos (u))

Apply the distributive law:

a (b + c) = ab + a c

a = cos (u), b = 1, c = cos (u)

= cos (u) \cdot 1 + cos (u) cos (u)

= 1 \cdot cos (u) + cos (u) cos (u)

Simplify

1 \cdot cos (u) + cos (u) cos (u) : cos (u) + cos^{2} (u)

1 \cdot cos (u) + cos (u) cos (u)

1 \cdot cos (u) = cos (u)

1 \cdot cos (u)

Multiply:

1 \cdot cos (u) = cos (u)

= cos (u)

cos (u) cos (u) = cos^{2} (u)

cos (u) cos (u)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos (u) cos (u) = cos^{1 + 1} (u)

= cos^{1 + 1} (u)

Add the numbers:

1 + 1 = 2

= cos^{2} (u)

= cos (u) + cos^{2} (u)

= cos (u) + cos^{2} (u)

= cos (u) + cos^{2} (u) - (1 + cos (u))

- (1 + cos (u)) : - 1 - cos (u)

- (1 + cos (u))

Distribute parentheses

= - (1) - (cos (u))

Apply minus-plus rules

+ (- a) = - a

= - 1 - cos (u)

= cos (u) + cos^{2} (u) - 1 - cos (u)

Simplify

cos (u) + cos^{2} (u) - 1 - cos (u) : cos^{2} (u) - 1

cos (u) + cos^{2} (u) - 1 - cos (u)

Group like terms

= cos (u) + cos^{2} (u) - cos (u) - 1

Add similar elements:

cos (u) - cos (u) = 0

= cos^{2} (u) - 1

= cos^{2} (u) - 1

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

Convert element to fraction:

sin (u) = \frac{sin ( u ) sin ( u )}{sin ( u )}

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

Since the denominators are equal, combine the fractions:

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

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

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

- sin (u) sin (u) + cos^{2} (u) - 1

sin (u) sin (u) = sin^{2} (u)

sin (u) sin (u)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (u) sin (u) = sin^{1 + 1} (u)

= sin^{1 + 1} (u)

Add the numbers:

1 + 1 = 2

= sin^{2} (u)

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

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

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

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

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

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

Rewrite using trig identities

- 1 + cos^{2} (u) - sin^{2} (u)

Use the Double Angle identity:

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

= - 1 + cos (2 u)

- 1 + cos (2 u) = 0

Move

1

to the right side

- 1 + cos (2 u) = 0

Add

1

to both sides

- 1 + cos (2 u) + 1 = 0 + 1

Simplify

cos (2 u) = 1

cos (2 u) = 1

General solutions for

cos (2 u) = 1

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

2 u = 0 + 2 πn

2 u = 0 + 2 πn

Solve

2 u = 0 + 2 πn : u = πn

2 u = 0 + 2 πn

0 + 2 πn = 2 πn

2 u = 2 πn

Divide both sides by

2

2 u = 2 πn

Divide both sides by

2

\frac{2 u}{2} = \frac{2 πn}{2}

Simplify

u = πn

u = πn

u = πn

Since the equation is undefined for:

πn

No Solution for u \in R

Graph

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Frequently Asked Questions (FAQ)

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