Is sec(x)-cos(x)=(1-cos(x))(1+sec(x)) ?

The answer to whether sec(x)-cos(x)=(1-cos(x))(1+sec(x)) is True

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prove sec(x)-cos(x)=(1-cos(x))(1+sec(x))

prove

sec (x) - cos (x) = (1 - cos (x)) (1 + sec (x))

True

Solution steps

sec (x) - cos (x) = (1 - cos (x)) (1 + sec (x))

Manipulating left side

sec (x) - cos (x)

Express with sin, cos

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

Manipulating right side

(1 - cos (x)) (1 + sec (x))

Express with sin, cos

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

Expand

(1 + cos (x)) (1 - cos (x)) : 1 - cos^{2} (x)

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

We showed that the two sides could take the same form

\Rightarrow True

(sec^{2} (x) - 1) (sin^{2} (x) - 1) = - sin^{2} (x)

csc^{2} (x) - csc^{2} (x) \cdot cos^{2} (x) = 1

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

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

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

Is sec(x)-cos(x)=(1-cos(x))(1+sec(x)) ?
The answer to whether sec(x)-cos(x)=(1-cos(x))(1+sec(x)) is True