What is the general solution for (sin(x)tan(x))/(1+cos(x))=tan(x) ?

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

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(sin(x)tan(x))/(1+cos(x))=tan(x)

Solution

\frac{sin ( x ) tan ( x )}{1 + cos ( x )} = tan (x)

Solution

No Solution for x \in R

Solution steps

\frac{sin ( x ) tan ( x )}{1 + cos ( x )} = tan (x)

Subtract

tan (x)

from both sides

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

Simplify

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

\frac{sin ( x ) tan ( x )}{1 + cos ( x )} - tan (x)

Convert element to fraction:

tan (x) = \frac{tan ( x ) ( 1 + cos ( x ) )}{1 + cos ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Factor

sin (x) tan (x) - tan (x) (1 + cos (x)) : tan (x) (sin (x) - 1 - cos (x))

sin (x) tan (x) - tan (x) (1 + cos (x))

Factor out common term

tan (x)

= tan (x) (sin (x) - (1 + cos (x)))

Expand

sin (x) - (cos (x) + 1) : sin (x) - 1 - cos (x)

sin (x) - (1 + cos (x))

- (1 + cos (x)) : - 1 - cos (x)

- (1 + cos (x))

Distribute parentheses

= - (1) - (cos (x))

Apply minus-plus rules

+ (- a) = - a

= - 1 - cos (x)

= sin (x) - 1 - cos (x)

= tan (x) (sin (x) - cos (x) - 1)

tan (x) (sin (x) - 1 - cos (x)) = 0

Solving each part separately

tan (x) = 0 or sin (x) - 1 - cos (x) = 0

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

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

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

Rewrite using trig identities

sin (x) - 1 - cos (x)

sin (x) - cos (x) = 2 sin (x - \frac{π}{4})

sin (x) - cos (x)

Rewrite as

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

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (x) - sin (\frac{π}{4}) cos (x))

Use the Angle Sum identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= 2 sin (x - \frac{π}{4})

= - 1 + 2 sin (x - \frac{π}{4})

- 1 + 2 sin (x - \frac{π}{4}) = 0

Move

1

to the right side

- 1 + 2 sin (x - \frac{π}{4}) = 0

Add

1

to both sides

- 1 + 2 sin (x - \frac{π}{4}) + 1 = 0 + 1

Simplify

2 sin (x - \frac{π}{4}) = 1

2 sin (x - \frac{π}{4}) = 1

Divide both sides by

2

2 sin (x - \frac{π}{4}) = 1

Divide both sides by

2

\frac{2 sin ( x - \frac{π}{4} )}{2} = \frac{1}{2}

Simplify

\frac{2 sin ( x - \frac{π}{4} )}{2} = \frac{1}{2}

Simplify

\frac{2 sin ( x - \frac{π}{4} )}{2} : sin (x - \frac{π}{4})

\frac{2 sin ( x - \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (x - \frac{π}{4})

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

sin (x - \frac{π}{4}) = \frac{2}{2}

sin (x - \frac{π}{4}) = \frac{2}{2}

sin (x - \frac{π}{4}) = \frac{2}{2}

General solutions for

sin (x - \frac{π}{4}) = \frac{2}{2}

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}{4} = \frac{π}{4} + 2 πn, x - \frac{π}{4} = \frac{3 π}{4} + 2 πn

x - \frac{π}{4} = \frac{π}{4} + 2 πn, x - \frac{π}{4} = \frac{3 π}{4} + 2 πn

Solve

x - \frac{π}{4} = \frac{π}{4} + 2 πn : x = \frac{π}{2} + 2 πn

x - \frac{π}{4} = \frac{π}{4} + 2 πn

Move

\frac{π}{4}

to the right side

x - \frac{π}{4} = \frac{π}{4} + 2 πn

Add

\frac{π}{4}

to both sides

x - \frac{π}{4} + \frac{π}{4} = \frac{π}{4} + 2 πn + \frac{π}{4}

Simplify

x - \frac{π}{4} + \frac{π}{4} = \frac{π}{4} + 2 πn + \frac{π}{4}

Simplify

x - \frac{π}{4} + \frac{π}{4} : x

x - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= x

Simplify

\frac{π}{4} + 2 πn + \frac{π}{4} : \frac{π}{2} + 2 πn

\frac{π}{4} + 2 πn + \frac{π}{4}

Group like terms

= \frac{π}{4} + \frac{π}{4} + 2 πn

Combine the fractions

\frac{π}{4} + \frac{π}{4} : \frac{π}{2}

Apply rule

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

= \frac{π + π}{4}

Add similar elements:

π + π = 2 π

= \frac{2 π}{4}

Cancel the common factor:

2

= \frac{π}{2}

= \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Solve

x - \frac{π}{4} = \frac{3 π}{4} + 2 πn : x = 2 πn + π

x - \frac{π}{4} = \frac{3 π}{4} + 2 πn

Move

\frac{π}{4}

to the right side

x - \frac{π}{4} = \frac{3 π}{4} + 2 πn

Add

\frac{π}{4}

to both sides

x - \frac{π}{4} + \frac{π}{4} = \frac{3 π}{4} + 2 πn + \frac{π}{4}

Simplify

x - \frac{π}{4} + \frac{π}{4} = \frac{3 π}{4} + 2 πn + \frac{π}{4}

Simplify

x - \frac{π}{4} + \frac{π}{4} : x

x - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= x

Simplify

\frac{3 π}{4} + 2 πn + \frac{π}{4} : 2 πn + π

\frac{3 π}{4} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{4} + \frac{3 π}{4}

Combine the fractions

\frac{π}{4} + \frac{3 π}{4} : π

Apply rule

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

= \frac{π + 3 π}{4}

Add similar elements:

π + 3 π = 4 π

= \frac{4 π}{4}

Divide the numbers:

\frac{4}{4} = 1

= π

= 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = \frac{π}{2} + 2 πn, x = 2 πn + π

Combine all the solutions

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

Since the equation is undefined for:

πn, \frac{π}{2} + 2 πn, 2 πn + π

No Solution for x \in R

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Frequently Asked Questions (FAQ)

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