Is (1+sin(-x))/(cos(x)tan(x)-1)=-1 ?

The answer to whether (1+sin(-x))/(cos(x)tan(x)-1)=-1 is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

prove

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

True

Solution steps

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

Manipulating left side

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

Use the negative angle identity:

sin (- x) = - sin (x)

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

= \frac{1 - sin ( x )}{- 1 + cos ( x ) \frac{s i n ( x )}{c o s ( x )}}

\frac{1 - sin ( x )}{- 1 + cos ( x ) \frac{s i n ( x )}{c o s ( x )}} = - 1

\frac{1 - sin ( x )}{- 1 + cos ( x ) \frac{s i n ( x )}{c o s ( 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)

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

Factor out

- 1

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

Refine

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

Apply rule

\frac{a}{a} = 1

= - 1

= - 1

= - 1

We showed that the two sides could take the same form

\Rightarrow True

Is (1+sin(-x))/(cos(x)tan(x)-1)=-1 ?
The answer to whether (1+sin(-x))/(cos(x)tan(x)-1)=-1 is True