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

The general solution for (cos(x)-sin(x))/(cos(x))=tan(x)-1 is x= pi/4+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

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

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

tan (x) - 1

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Add similar elements:

cos (x) + 1 \cdot cos (x) = 2 cos (x)

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

\frac{2 cos ( x ) - sin ( x ) - tan ( x ) cos ( x )}{cos ( x )} = 0

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

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

Rewrite using trig identities

- sin (x) + 2 cos (x) - cos (x) tan (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x)

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

Add similar elements:

- sin (x) - sin (x) = - 2 sin (x)

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

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

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

2 - 2 tan (x) = 0

2 - 2 tan (x) = 0

Move

2

to the right side

2 - 2 tan (x) = 0

Subtract

2

from both sides

2 - 2 tan (x) - 2 = 0 - 2

Simplify

- 2 tan (x) = - 2

- 2 tan (x) = - 2

Divide both sides by

- 2

- 2 tan (x) = - 2

Divide both sides by

- 2

\frac{- 2 tan ( x )}{- 2} = \frac{- 2}{- 2}

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

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 = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Popular Examples

cos(φ)= 1/4

cos (φ) = \frac{1}{4}

-2cos(x)=sin(x)

- 2 cos (x) = sin (x)

sin(x)=-0.9781

sin (x) = - 0.9781

tan(x)=3.73

tan (x) = 3.73

tan(θ)=-2/5 ,sin(θ)>0

tan (θ) = - \frac{2}{5}, sin (θ) > 0

Frequently Asked Questions (FAQ)

What is the general solution for (cos(x)-sin(x))/(cos(x))=tan(x)-1 ?
The general solution for (cos(x)-sin(x))/(cos(x))=tan(x)-1 is x= pi/4+pin