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

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

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

prove

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

True

Solution steps

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

Manipulating left side

\frac{1 - cot ( a )}{csc ( a )}

Express with sin, cos

\frac{1 - cot ( a )}{csc ( a )}

Use the basic trigonometric identity:

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

= \frac{1 - \frac{c o s ( a )}{s i n ( a )}}{csc ( a )}

Use the basic trigonometric identity:

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

= \frac{1 - \frac{c o s ( a )}{s i n ( a )}}{\frac{1}{s i n ( a )}}

Simplify

\frac{1 - \frac{c o s ( a )}{s i n ( a )}}{\frac{1}{s i n ( a )}} : sin (a) - cos (a)

\frac{1 - \frac{c o s ( a )}{s i n ( a )}}{\frac{1}{s i n ( a )}}

Apply the fraction rule:

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

= \frac{( 1 - \frac{c o s ( a )}{s i n ( a )} ) sin ( a )}{1}

Join

1 - \frac{cos ( a )}{sin ( a )} : \frac{sin ( a ) - cos ( a )}{sin ( a )}

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

Convert element to fraction:

1 = \frac{1 sin ( a )}{sin ( a )}

= \frac{1 \cdot sin ( a )}{sin ( a )} - \frac{cos ( a )}{sin ( a )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot sin ( a ) - cos ( a )}{sin ( a )}

Multiply:

1 \cdot sin (a) = sin (a)

= \frac{sin ( a ) - cos ( a )}{sin ( a )}

= \frac{\frac{s i n ( a ) - c o s ( a )}{s i n ( a )} sin ( a )}{1}

Apply the fraction rule:

\frac{a}{1} = a

= \frac{sin ( a ) - cos ( a )}{sin ( a )} sin (a)

Multiply fractions:

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

= \frac{( sin ( a ) - cos ( a ) ) sin ( a )}{sin ( a )}

Cancel the common factor:

sin (a)

= sin (a) - cos (a)

= sin (a) - cos (a)

= - cos (a) + sin (a)

= sin (a) - cos (a)

We showed that the two sides could take the same form

\Rightarrow True

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