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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Multiply by

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

= \frac{sin ( x )}{\frac{( 1 - c o s ( x ) ) ( 1 + c o s ( x ) )}{1 + c o s ( x )}} - \frac{1}{sin ( x )}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(1 - cos (x)) (1 + cos (x)) = 1 - cos^{2} (x)

= \frac{sin ( x )}{\frac{1 - c o s ^{2} ( x )}{1 + c o s ( x )}} - \frac{1}{sin ( x )}

Use the Pythagorean identity:

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

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

= \frac{sin ( x )}{\frac{s i n ^{2} ( x )}{1 + c o s ( x )}} - \frac{1}{sin ( x )}

Simplify

\frac{sin ( x )}{\frac{s i n ^{2} ( x )}{1 + c o s ( x )}} - \frac{1}{sin ( x )} : \frac{cos ( x )}{sin ( x )}

\frac{sin ( x )}{\frac{s i n ^{2} ( x )}{1 + c o s ( x )}} - \frac{1}{sin ( x )}

\frac{sin ( x )}{\frac{s i n ^{2} ( x )}{1 + c o s ( x )}} = \frac{1 + cos ( x )}{sin ( x )}

\frac{sin ( x )}{\frac{s i n ^{2} ( x )}{1 + c o s ( x )}}

Apply the fraction rule:

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

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

Cancel the common factor:

sin (x)

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

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

Apply rule

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

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

1 + cos (x) - 1 = cos (x)

1 + cos (x) - 1

Group like terms

= cos (x) + 1 - 1

1 - 1 = 0

= cos (x)

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

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

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

Use the basic trigonometric identity:

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

= cot (x)

We showed that the two sides could take the same form

\Rightarrow True

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