Is 1/(1-cos(x))-1/(1+cos(x))=2cot(x)csc(x) ?

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

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prove 1/(1-cos(x))-1/(1+cos(x))=2cot(x)csc(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Simplify

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

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

Rewrite using trig identities

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

Manipulating right side

2 cot (x) csc (x)

Express with sin, cos

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

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

(1 + cot^{2} (x)) sin^{2} (x) = 1

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