Is (1+cos(x))/(1-cos(x))-(1-cos(x))/(1+cos(x))=4cot(x)csc(x) ?

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

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prove (1+cos(x))/(1-cos(x))-(1-cos(x))/(1+cos(x))=4cot(x)csc(x)

prove

\frac{1 + cos ( x )}{1 - cos ( x )} - \frac{1 - cos ( x )}{1 + cos ( x )} = 4 cot (x) csc (x)

True

Solution steps

\frac{1 + cos ( x )}{1 - cos ( x )} - \frac{1 - cos ( x )}{1 + cos ( x )} = 4 cot (x) csc (x)

Manipulating left side

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

Simplify

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

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

Rewrite using trig identities

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

Manipulating right side

4 cot (x) csc (x)

Express with sin, cos

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{1 - tan ^{2} ( x )}{cot ^{2} ( x ) - 1} = tan^{2} (x)

\frac{2 tan ( θ )}{1 + tan ^{2} ( θ )} = sin (2 θ)

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

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

cos (z + w) = cos (z) cos (w) - sin (z) sin (w)

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