Is cos(x)(sec(x)-1)=1-cos(x) ?

The answer to whether cos(x)(sec(x)-1)=1-cos(x) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

prove cos(x)(sec(x)-1)=1-cos(x)

prove

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

True

Solution steps

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

Manipulating left side

cos (x) (sec (x) - 1)

Express with sin, cos

(- 1 + sec (x)) cos (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= - - cos (x) + 1

= - cos (x) + 1

= 1 - cos (x)

We showed that the two sides could take the same form

\Rightarrow True

Is cos(x)(sec(x)-1)=1-cos(x) ?
The answer to whether cos(x)(sec(x)-1)=1-cos(x) is True