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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

- 1 + \frac{1}{cos ( t )}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (t) = cos (t)

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

- cos (t) + 1

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sec (t)

sec (t)

sec (t)

We showed that the two sides could take the same form

\Rightarrow True

tan (\frac{7 π}{6}) = \frac{3}{3}

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

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

cot (θ) \cdot sec (θ) = csc (θ)

csc^{2} (x) + cot^{2} (x) = 1 + 2 cot^{2} (x)

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