Is (4sec(θ)-4)/(1-cos(θ))=4sec(θ) ?

The answer to whether (4sec(θ)-4)/(1-cos(θ))=4sec(θ) is True

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prove (4sec(θ)-4)/(1-cos(θ))=4sec(θ)

prove

\frac{4 sec ( θ ) - 4}{1 - cos ( θ )} = 4 sec (θ)

True

Solution steps

\frac{4 sec ( θ ) - 4}{1 - cos ( θ )} = 4 sec (θ)

Manipulating left side

\frac{4 sec ( θ ) - 4}{1 - cos ( θ )}

Express with sin, cos

\frac{- 4 + 4 sec ( θ )}{1 - cos ( θ )}

Use the basic trigonometric identity:

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

= \frac{- 4 + 4 \cdot \frac{1}{c o s ( θ )}}{1 - cos ( θ )}

Simplify

\frac{- 4 + 4 \cdot \frac{1}{c o s ( θ )}}{1 - cos ( θ )} : \frac{4}{cos ( θ )}

\frac{- 4 + 4 \cdot \frac{1}{c o s ( θ )}}{1 - cos ( θ )}

4 \cdot \frac{1}{cos ( θ )} = \frac{4}{cos ( θ )}

4 \cdot \frac{1}{cos ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 4}{cos ( θ )}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{cos ( θ )}

= \frac{- 4 + \frac{4}{c o s ( θ )}}{1 - cos ( θ )}

Join

- 4 + \frac{4}{cos ( θ )} : \frac{- 4 cos ( θ ) + 4}{cos ( θ )}

- 4 + \frac{4}{cos ( θ )}

Convert element to fraction:

4 = \frac{4 cos ( θ )}{cos ( θ )}

= - \frac{4 cos ( θ )}{cos ( θ )} + \frac{4}{cos ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 4 cos ( θ ) + 4}{cos ( θ )}

= \frac{\frac{- 4 c o s ( θ ) + 4}{c o s ( θ )}}{1 - cos ( θ )}

Apply the fraction rule:

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

= \frac{- 4 cos ( θ ) + 4}{cos ( θ ) ( 1 - cos ( θ ) )}

Factor

- 4 cos (θ) + 4 : 4 (- cos (θ) + 1)

- 4 cos (θ) + 4

Rewrite as

= - 4 cos (θ) + 4 \cdot 1

Factor out common term

4

= 4 (- cos (θ) + 1)

= \frac{4 ( - cos ( θ ) + 1 )}{cos ( θ ) ( 1 - cos ( θ ) )}

Cancel the common factor:

- cos (θ) + 1

= \frac{4}{cos ( θ )}

= \frac{4}{cos ( θ )}

= \frac{4}{cos ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{4}{\frac{1}{s e c ( θ )}}

Simplify

\frac{4}{\frac{1}{s e c ( θ )}}

Apply the fraction rule:

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

= \frac{4 sec ( θ )}{1}

Apply rule

\frac{a}{1} = a

= 4 sec (θ)

4 sec (θ)

4 sec (θ)

We showed that the two sides could take the same form

\Rightarrow True

tan^{2} (θ) = \frac{1 - cos ^{2} ( θ )}{cos ^{2} ( θ )}

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

tan (u) - 1 tan (u) + 1 = 1 - cot (u) 1 + cot (u)

cos (x) = sec (x) - sin (x) tan (x)

arccsc (x) = sin (x)

Is (4sec(θ)-4)/(1-cos(θ))=4sec(θ) ?
The answer to whether (4sec(θ)-4)/(1-cos(θ))=4sec(θ) is True