Is (7sec(θ)-7)/(1-cos(θ))=7sec(θ) ?

The answer to whether (7sec(θ)-7)/(1-cos(θ))=7sec(θ) is True

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prove (7sec(θ)-7)/(1-cos(θ))=7sec(θ)

prove

\frac{7 sec ( θ ) - 7}{1 - cos ( θ )} = 7 sec (θ)

True

Solution steps

\frac{7 sec ( θ ) - 7}{1 - cos ( θ )} = 7 sec (θ)

Manipulating left side

\frac{7 sec ( θ ) - 7}{1 - cos ( θ )}

Express with sin, cos

\frac{- 7 + 7 sec ( θ )}{1 - cos ( θ )}

Use the basic trigonometric identity:

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

= \frac{- 7 + 7 \cdot \frac{1}{c o s ( θ )}}{1 - cos ( θ )}

Simplify

\frac{- 7 + 7 \cdot \frac{1}{c o s ( θ )}}{1 - cos ( θ )} : \frac{7}{cos ( θ )}

\frac{- 7 + 7 \cdot \frac{1}{c o s ( θ )}}{1 - cos ( θ )}

7 \cdot \frac{1}{cos ( θ )} = \frac{7}{cos ( θ )}

7 \cdot \frac{1}{cos ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 7}{cos ( θ )}

Multiply the numbers:

1 \cdot 7 = 7

= \frac{7}{cos ( θ )}

= \frac{- 7 + \frac{7}{c o s ( θ )}}{1 - cos ( θ )}

Join

- 7 + \frac{7}{cos ( θ )} : \frac{- 7 cos ( θ ) + 7}{cos ( θ )}

- 7 + \frac{7}{cos ( θ )}

Convert element to fraction:

7 = \frac{7 cos ( θ )}{cos ( θ )}

= - \frac{7 cos ( θ )}{cos ( θ )} + \frac{7}{cos ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 7 cos ( θ ) + 7}{cos ( θ )}

= \frac{\frac{- 7 c o s ( θ ) + 7}{c o s ( θ )}}{1 - cos ( θ )}

Apply the fraction rule:

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

= \frac{- 7 cos ( θ ) + 7}{cos ( θ ) ( 1 - cos ( θ ) )}

Factor

- 7 cos (θ) + 7 : 7 (- cos (θ) + 1)

- 7 cos (θ) + 7

Rewrite as

= - 7 cos (θ) + 7 \cdot 1

Factor out common term

7

= 7 (- cos (θ) + 1)

= \frac{7 ( - cos ( θ ) + 1 )}{cos ( θ ) ( 1 - cos ( θ ) )}

Cancel the common factor:

- cos (θ) + 1

= \frac{7}{cos ( θ )}

= \frac{7}{cos ( θ )}

= \frac{7}{cos ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{7}{\frac{1}{s e c ( θ )}}

Simplify

\frac{7}{\frac{1}{s e c ( θ )}}

Apply the fraction rule:

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

= \frac{7 sec ( θ )}{1}

Apply rule

\frac{a}{1} = a

= 7 sec (θ)

7 sec (θ)

7 sec (θ)

We showed that the two sides could take the same form

\Rightarrow True

Is (7sec(θ)-7)/(1-cos(θ))=7sec(θ) ?
The answer to whether (7sec(θ)-7)/(1-cos(θ))=7sec(θ) is True