Is (sec(α))/(1-cos(α))=(sec(α)+1)/(sin^2(α)) ?

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

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prove (sec(α))/(1-cos(α))=(sec(α)+1)/(sin^2(α))

prove

\frac{sec ( α )}{1 - cos ( α )} = \frac{sec ( α ) + 1}{sin ^{2} ( α )}

True

Solution steps

\frac{sec ( α )}{1 - cos ( α )} = \frac{sec ( α ) + 1}{sin ^{2} ( α )}

Manipulating left side

\frac{sec ( α )}{1 - cos ( α )}

Rewrite using trig identities

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

Expand

\frac{( 1 + cos ( α ) ) sec ( α )}{sin ^{2} ( α )} : \frac{sec ( α )}{sin ^{2} ( α )} + \frac{sec ( α ) cos ( α )}{sin ^{2} ( α )}

= \frac{cos ( α ) sec ( α )}{sin ^{2} ( α )} + \frac{sec ( α )}{sin ^{2} ( α )}

Rewrite using trig identities

= \frac{1 + sec ( α )}{sin ^{2} ( α )}

= \frac{sec ( α ) + 1}{sin ^{2} ( α )}

We showed that the two sides could take the same form

\Rightarrow True

2 csc (2 x) = csc (x) sec (x)

cos (\frac{π}{4}) = \frac{2}{2}

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

sec^{2} (x) - tan (x) cot (x) = tan^{2} (x)

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

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