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

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

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prove (1-cos(x))/(sin(x))=csc(x)-cot(x)

prove

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

True

Solution steps

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

Manipulating right side

csc (x) - cot (x)

Express with sin, cos

- cot (x) + csc (x)

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= - \frac{cos ( x )}{sin ( x )} + csc (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

cos^{2} (β) - sin^{2} (β) = 1 - 2 sin^{2} (β)

csc (2 x) = \frac{1}{2} sec (x) csc (x)

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

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

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

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