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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Expand

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

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

Rewrite using trig identities

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

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

We showed that the two sides could take the same form

\Rightarrow True

(1 - sin (x)) (1 + csc (x)) = csc (x) - sin (x)

1 - cos^{2} (θ) = sin^{2} (θ)

tan (\frac{π}{4} - θ) = \frac{cos ( θ ) - sin ( θ )}{cos ( θ ) + sin ( θ )}

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

csc (B) - sin (B) = cot (B) cos (B)

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