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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Simplify

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

1 + cot^{2} (θ) = \frac{sec ^{2} ( θ )}{sec ^{2} ( θ ) - 1}

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

cos^{2} (u) (sec^{2} (u) - 1) = sin^{2} (u)

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

(\frac{1}{2} sin (2 x)) (1 - cos (4 x)) = sin^{3} (2 x)

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