Is (tan(x)*cot(x))/(csc(x))=sin(x) ?

The answer to whether (tan(x)*cot(x))/(csc(x))=sin(x) is True

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prove (tan(x)*cot(x))/(csc(x))=sin(x)

prove

\frac{tan ( x ) \cdot cot ( x )}{csc ( x )} = sin (x)

True

Solution steps

\frac{tan ( x ) cot ( x )}{csc ( x )} = sin (x)

Manipulating left side

\frac{tan ( x ) cot ( x )}{csc ( x )}

Express with sin, cos

\frac{cot ( x ) tan ( x )}{csc ( x )}

Use the basic trigonometric identity:

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

= \frac{\frac{c o s ( x )}{s i n ( x )} tan ( x )}{csc ( x )}

Use the basic trigonometric identity:

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

= \frac{\frac{c o s ( x )}{s i n ( x )} \cdot \frac{s i n ( x )}{c o s ( x )}}{csc ( x )}

Use the basic trigonometric identity:

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

= \frac{\frac{c o s ( x )}{s i n ( x )} \cdot \frac{s i n ( x )}{c o s ( x )}}{\frac{1}{s i n ( x )}}

Simplify

\frac{\frac{c o s ( x )}{s i n ( x )} \cdot \frac{s i n ( x )}{c o s ( x )}}{\frac{1}{s i n ( x )}} : sin (x)

\frac{\frac{c o s ( x )}{s i n ( x )} \cdot \frac{s i n ( x )}{c o s ( x )}}{\frac{1}{s i n ( x )}}

Apply the fraction rule:

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

= \frac{\frac{c o s ( x )}{s i n ( x )} \cdot \frac{s i n ( x )}{c o s ( x )} sin ( x )}{1}

Apply the fraction rule:

\frac{a}{1} = a

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

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

Cancel the common factor:

sin (x)

= sin (x)

= sin (x)

= sin (x)

We showed that the two sides could take the same form

\Rightarrow True

Is (tan(x)*cot(x))/(csc(x))=sin(x) ?
The answer to whether (tan(x)*cot(x))/(csc(x))=sin(x) is True