Is ((tan^2(x)))/(sec(x)+1)=((1-cos(x)))/(cos(x)) ?

The answer to whether ((tan^2(x)))/(sec(x)+1)=((1-cos(x)))/(cos(x)) is True

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prove ((tan^2(x)))/(sec(x)+1)=((1-cos(x)))/(cos(x))

prove

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

True

Solution steps

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

Manipulating left side

\frac{tan ^{2} ( x )}{sec ( x ) + 1}

Express with sin, cos

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

Rewrite using trig identities

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

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

Simplify

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

We showed that the two sides could take the same form

\Rightarrow True

1 + sin (x) = \frac{csc ( x ) + 1}{csc ( x )}

1 + sin (θ) = csc (θ)

\frac{1 + sin ( x )}{1 - sin ( x )} - \frac{1 - sin ( x )}{1 + sin ( x )} = \frac{4 tan ( x )}{cos ( x )}

sin (3 x) = 4 sin (x) \cdot cos^{2} (x) - sin (x)

sin (θ) = \frac{4}{3} sin^{3} (θ) + \frac{1}{3} sin^{3} (θ)

Is ((tan^2(x)))/(sec(x)+1)=((1-cos(x)))/(cos(x)) ?
The answer to whether ((tan^2(x)))/(sec(x)+1)=((1-cos(x)))/(cos(x)) is True