Is (1+sin(x))/(1-sin(x))-(1-sin(x))/(1+sin(x))=(4tan(x))/(cos(x)) ?

The answer to whether (1+sin(x))/(1-sin(x))-(1-sin(x))/(1+sin(x))=(4tan(x))/(cos(x)) is True

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prove (1+sin(x))/(1-sin(x))-(1-sin(x))/(1+sin(x))=(4tan(x))/(cos(x))

prove

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

True

Solution steps

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

Manipulating left side

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

Simplify

\frac{1 + sin ( x )}{1 - sin ( x )} - \frac{1 - sin ( x )}{1 + sin ( x )} : \frac{4 sin ( x )}{( - sin ( x ) + 1 ) ( sin ( x ) + 1 )}

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

Rewrite using trig identities

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

Rewrite using trig identities

\frac{4 tan ( x )}{cos ( x )}

We showed that the two sides could take the same form

\Rightarrow True

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)

sin (x - \frac{3 π}{2}) + sin (\frac{π}{2} + x) = 2 cos (x)

\frac{1}{cos ( x ) \cdot sin ( x )} - tan (x) = cot (x)

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

Is (1+sin(x))/(1-sin(x))-(1-sin(x))/(1+sin(x))=(4tan(x))/(cos(x)) ?
The answer to whether (1+sin(x))/(1-sin(x))-(1-sin(x))/(1+sin(x))=(4tan(x))/(cos(x)) is True