Is (1+sin(θ))/(1-sin(θ))-(1-sin(θ))/(1+sin(θ))=4tan(θ)sec(θ) ?

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

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prove (1+sin(θ))/(1-sin(θ))-(1-sin(θ))/(1+sin(θ))=4tan(θ)sec(θ)

prove

\frac{1 + sin ( θ )}{1 - sin ( θ )} - \frac{1 - sin ( θ )}{1 + sin ( θ )} = 4 tan (θ) sec (θ)

True

Solution steps

\frac{1 + sin ( θ )}{1 - sin ( θ )} - \frac{1 - sin ( θ )}{1 + sin ( θ )} = 4 tan (θ) sec (θ)

Manipulating left side

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

Simplify

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

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

Rewrite using trig identities

= \frac{4 sin ( θ )}{cos ^{2} ( θ )}

Manipulating right side

4 tan (θ) sec (θ)

Express with sin, cos

= \frac{4 sin ( θ )}{cos ^{2} ( θ )}

We showed that the two sides could take the same form

\Rightarrow True

cos (7 x) + cos (3 x) = 2 cos (5 x) cos (2 x)

cot (- α) cos (- α) + sin (- α) = - csc (α)

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

sin (θ) cos (θ) tan (θ) = 1 - cos^{2} (θ)

cot^{2} (- θ) + 1 = csc^{2} (θ)

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