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

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

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

prove

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

True

Solution steps

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

Manipulating left side

\frac{1}{1 - sin ( x )} - \frac{1}{1 + sin ( x )}

Simplify

- \frac{1}{1 + sin ( x )} + \frac{1}{1 - sin ( x )} : \frac{2 sin ( x )}{( sin ( x ) + 1 ) ( - sin ( x ) + 1 )}

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

Rewrite using trig identities

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

Rewrite using trig identities

\frac{2 tan ( x )}{cos ( x )}

We showed that the two sides could take the same form

\Rightarrow True

cot (b) sin (b) = cos (b)

cos (t) cot (t) = \frac{1 - sin ^{2} ( t )}{sin ( t )}

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

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

tan^{2} (θ) - sin^{2} (θ) = sin^{2} (θ) tan^{2} (θ)

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