Is (cot^3(t))/(csc(t))=cos(t)cot^2(t) ?

The answer to whether (cot^3(t))/(csc(t))=cos(t)cot^2(t) is True

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prove (cot^3(t))/(csc(t))=cos(t)cot^2(t)

prove

\frac{cot ^{3} ( t )}{csc ( t )} = cos (t) cot^{2} (t)

True

Solution steps

\frac{cot ^{3} ( t )}{csc ( t )} = cos (t) cot^{2} (t)

Manipulating left side

\frac{cot ^{3} ( t )}{csc ( t )}

Express with sin, cos

\frac{cot ^{3} ( t )}{csc ( t )}

Use the basic trigonometric identity:

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

= \frac{( \frac{c o s ( t )}{s i n ( t )} ) ^{3}}{csc ( t )}

Use the basic trigonometric identity:

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

= \frac{( \frac{c o s ( t )}{s i n ( t )} ) ^{3}}{\frac{1}{s i n ( t )}}

Simplify

\frac{( \frac{c o s ( t )}{s i n ( t )} ) ^{3}}{\frac{1}{s i n ( t )}} : \frac{cos ^{3} ( t )}{sin ^{2} ( t )}

\frac{( \frac{c o s ( t )}{s i n ( t )} ) ^{3}}{\frac{1}{s i n ( t )}}

Apply the fraction rule:

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

= \frac{( \frac{c o s ( t )}{s i n ( t )} ) ^{3} sin ( t )}{1}

(\frac{cos ( t )}{sin ( t )})^{3} = \frac{cos ^{3} ( t )}{sin ^{3} ( t )}

(\frac{cos ( t )}{sin ( t )})^{3}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{cos ^{3} ( t )}{sin ^{3} ( t )}

= \frac{\frac{c o s ^{3} ( t )}{s i n ^{3} ( t )} sin ( t )}{1}

Multiply

\frac{cos ^{3} ( t )}{sin ^{3} ( t )} sin (t) : \frac{cos ^{3} ( t )}{sin ^{2} ( t )}

\frac{cos ^{3} ( t )}{sin ^{3} ( t )} sin (t)

Multiply fractions:

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

= \frac{cos ^{3} ( t ) sin ( t )}{sin ^{3} ( t )}

Cancel the common factor:

sin (t)

= \frac{cos ^{3} ( t )}{sin ^{2} ( t )}

= \frac{\frac{c o s ^{3} ( t )}{s i n ^{2} ( t )}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= \frac{cos ^{3} ( t )}{sin ^{2} ( t )}

= \frac{cos ^{3} ( t )}{sin ^{2} ( t )}

= \frac{cos ^{3} ( t )}{sin ^{2} ( t )}

Rewrite using trig identities

= \frac{cos ^{2} ( t )}{sin ( t )} \cdot \frac{cos ( t )}{sin ( t )}

Use the basic trigonometric identity:

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

\frac{cos ^{2} ( t ) cot ( t )}{sin ( t )}

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

Use the basic trigonometric identity:

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

cos (t) cot (t) cot (t)

Simplify

cos (t) cot (t) cot (t) : cot^{2} (t) cos (t)

cos (t) cot (t) cot (t)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cot (t) cot (t) = cot^{1 + 1} (t)

= cos (t) cot^{1 + 1} (t)

Add the numbers:

1 + 1 = 2

= cos (t) cot^{2} (t)

cot^{2} (t) cos (t)

cot^{2} (t) cos (t)

= cos (t) cot^{2} (t)

We showed that the two sides could take the same form

\Rightarrow True

Is (cot^3(t))/(csc(t))=cos(t)cot^2(t) ?
The answer to whether (cot^3(t))/(csc(t))=cos(t)cot^2(t) is True