Is (sec(θ)-1)/(1-cos(θ))=sec(θ) ?

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

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prove (sec(θ)-1)/(1-cos(θ))=sec(θ)

prove

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

True

Solution steps

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

Manipulating left side

\frac{sec ( θ ) - 1}{1 - cos ( θ )}

Express with sin, cos

\frac{- 1 + sec ( θ )}{1 - cos ( θ )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

- 1 + \frac{1}{cos ( θ )}

Convert element to fraction:

1 = \frac{1 cos ( θ )}{cos ( θ )}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (θ) = cos (θ)

= \frac{- cos ( θ ) + 1}{cos ( θ )}

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

Apply the fraction rule:

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

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

Cancel the common factor:

- cos (θ) + 1

= \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{s e c ( θ )}}

Simplify

\frac{1}{\frac{1}{s e c ( θ )}}

Apply the fraction rule:

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

= \frac{sec ( θ )}{1}

Apply rule

\frac{a}{1} = a

= sec (θ)

sec (θ)

sec (θ)

We showed that the two sides could take the same form

\Rightarrow True

cot (θ) + tan (θ) = sec (θ) csc (θ)

cos (\frac{π}{2} - θ) = sin (θ)

cos (x) cot (x) + sin (x) = csc (x)

sin (θ) tan (θ) + cos (θ) = sec (θ)

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

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