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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

- cos (x) + 1

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

Apply rule

\frac{a}{1} = a

= sec (x)

sec (x)

sec (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (x + 2 π) = sin (x)

cot^{2} (x) cos^{2} (x) + cos^{2} (x) = cot^{2} (x)

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

sin (a) sec (a) cot (a) = 1

3 + 3 cos (x) = 2 (1 - cos^{2} (x))

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