Is cot(x)-1=(csc(x)-sec(x))/(sec(x)) ?

The answer to whether cot(x)-1=(csc(x)-sec(x))/(sec(x)) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

cot (x) - 1

Express with sin, cos

- 1 + cot (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Convert element to fraction:

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

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

Multiply:

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

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{cos ( x ) - \frac{1}{c s c ( x )}}{\frac{1}{c s c ( x )}}

Use the basic trigonometric identity:

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

\frac{\frac{1}{s e c ( x )} - \frac{1}{c s c ( x )}}{\frac{1}{c s c ( x )}}

Simplify

\frac{\frac{1}{s e c ( x )} - \frac{1}{c s c ( x )}}{\frac{1}{c s c ( x )}}

Apply the fraction rule:

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

= \frac{( \frac{1}{s e c ( x )} - \frac{1}{c s c ( x )} ) csc ( x )}{1}

Join

\frac{1}{sec ( x )} - \frac{1}{csc ( x )} : \frac{csc ( x ) - sec ( x )}{sec ( x ) csc ( x )}

\frac{1}{sec ( x )} - \frac{1}{csc ( x )}

Least Common Multiplier of

sec (x), csc (x) : sec (x) csc (x)

sec (x), csc (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sec (x)

csc (x)

= sec (x) csc (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sec (x) csc (x)

For

\frac{1}{sec ( x )} :

multiply the denominator and numerator by

csc (x)

\frac{1}{sec ( x )} = \frac{1 \cdot csc ( x )}{sec ( x ) csc ( x )} = \frac{csc ( x )}{sec ( x ) csc ( x )}

For

\frac{1}{csc ( x )} :

multiply the denominator and numerator by

sec (x)

\frac{1}{csc ( x )} = \frac{1 \cdot sec ( x )}{csc ( x ) sec ( x )} = \frac{sec ( x )}{sec ( x ) csc ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

= \frac{\frac{c s c ( x ) - s e c ( x )}{s e c ( x ) c s c ( x )} csc ( x )}{1}

Apply the fraction rule:

\frac{a}{1} = a

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

Multiply fractions:

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

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

Cancel the common factor:

csc (x)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

Popular Examples

prove tan(θ)=2 prove

tan (θ) = 2

prove sec(x)=sec(-x)prove

sec (x) = sec (- x)

prove csc^2(θ)-1=cot^2(θ)prove

csc^{2} (θ) - 1 = cot^{2} (θ)

prove ((cot(a)))/((csc(a)))=cos(a)prove

\frac{( cot ( a ) )}{( csc ( a ) )} = cos (a)

prove cos(pi/(12))=(sqrt(2+\sqrt{3)})/2 prove

cos (\frac{π}{12}) = \frac{2 + 3}{2}

Frequently Asked Questions (FAQ)

Is cot(x)-1=(csc(x)-sec(x))/(sec(x)) ?
The answer to whether cot(x)-1=(csc(x)-sec(x))/(sec(x)) is True