Is (3sec(θ)-3)/(1-cos(θ))=3sec(θ) ?

The answer to whether (3sec(θ)-3)/(1-cos(θ))=3sec(θ) is True

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prove (3sec(θ)-3)/(1-cos(θ))=3sec(θ)

prove

\frac{3 sec ( θ ) - 3}{1 - cos ( θ )} = 3 sec (θ)

True

Solution steps

\frac{3 sec ( θ ) - 3}{1 - cos ( θ )} = 3 sec (θ)

Manipulating left side

\frac{3 sec ( θ ) - 3}{1 - cos ( θ )}

Express with sin, cos

\frac{- 3 + 3 sec ( θ )}{1 - cos ( θ )}

Use the basic trigonometric identity:

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

= \frac{- 3 + 3 \cdot \frac{1}{c o s ( θ )}}{1 - cos ( θ )}

Simplify

\frac{- 3 + 3 \cdot \frac{1}{c o s ( θ )}}{1 - cos ( θ )} : \frac{3}{cos ( θ )}

\frac{- 3 + 3 \cdot \frac{1}{c o s ( θ )}}{1 - cos ( θ )}

3 \cdot \frac{1}{cos ( θ )} = \frac{3}{cos ( θ )}

3 \cdot \frac{1}{cos ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 3}{cos ( θ )}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{cos ( θ )}

= \frac{- 3 + \frac{3}{c o s ( θ )}}{1 - cos ( θ )}

Join

- 3 + \frac{3}{cos ( θ )} : \frac{- 3 cos ( θ ) + 3}{cos ( θ )}

- 3 + \frac{3}{cos ( θ )}

Convert element to fraction:

3 = \frac{3 cos ( θ )}{cos ( θ )}

= - \frac{3 cos ( θ )}{cos ( θ )} + \frac{3}{cos ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 3 cos ( θ ) + 3}{cos ( θ )}

= \frac{\frac{- 3 c o s ( θ ) + 3}{c o s ( θ )}}{1 - cos ( θ )}

Apply the fraction rule:

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

= \frac{- 3 cos ( θ ) + 3}{cos ( θ ) ( 1 - cos ( θ ) )}

Factor

- 3 cos (θ) + 3 : 3 (- cos (θ) + 1)

- 3 cos (θ) + 3

Rewrite as

= - 3 cos (θ) + 3 \cdot 1

Factor out common term

3

= 3 (- cos (θ) + 1)

= \frac{3 ( - cos ( θ ) + 1 )}{cos ( θ ) ( 1 - cos ( θ ) )}

Cancel the common factor:

- cos (θ) + 1

= \frac{3}{cos ( θ )}

= \frac{3}{cos ( θ )}

= \frac{3}{cos ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{3}{\frac{1}{s e c ( θ )}}

Simplify

\frac{3}{\frac{1}{s e c ( θ )}}

Apply the fraction rule:

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

= \frac{3 sec ( θ )}{1}

Apply rule

\frac{a}{1} = a

= 3 sec (θ)

3 sec (θ)

3 sec (θ)

We showed that the two sides could take the same form

\Rightarrow True

cot (\frac{π}{2} - θ) = tan (θ)

cos (\frac{π}{2} - a) = sin (a)

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

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

sin^{2} (2 x) = \frac{1}{2} - \frac{1}{2} cos (4 x)

Is (3sec(θ)-3)/(1-cos(θ))=3sec(θ) ?
The answer to whether (3sec(θ)-3)/(1-cos(θ))=3sec(θ) is True