Is (csc^2(t))/(cot(t))=csc(t)sec(t) ?

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

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prove (csc^2(t))/(cot(t))=csc(t)sec(t)

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

\frac{csc ^{2} ( t )}{cot ( t )}

Express with sin, cos

\frac{csc ^{2} ( t )}{cot ( t )}

Use the basic trigonometric identity:

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

= \frac{( \frac{1}{s i n ( t )} ) ^{2}}{cot ( t )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

= \frac{( \frac{1}{s i n ( t )} ) ^{2} sin ( t )}{cos ( t )}

(\frac{1}{sin ( t )})^{2} = \frac{1}{sin ^{2} ( t )}

(\frac{1}{sin ( t )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{sin ^{2} ( t )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( t )}

= \frac{\frac{1}{s i n ^{2} ( t )} sin ( t )}{cos ( t )}

Multiply

\frac{1}{sin ^{2} ( t )} sin (t) : \frac{1}{sin ( t )}

\frac{1}{sin ^{2} ( t )} sin (t)

Multiply fractions:

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

= \frac{1 \cdot sin ( t )}{sin ^{2} ( t )}

Multiply:

1 \cdot sin (t) = sin (t)

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

Cancel the common factor:

sin (t)

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

= \frac{\frac{1}{s i n ( t )}}{cos ( t )}

Apply the fraction rule:

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

= \frac{1}{sin ( t ) cos ( t )}

= \frac{1}{sin ( t ) cos ( t )}

= \frac{1}{cos ( t ) sin ( t )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{cos ( t ) \frac{1}{c s c ( t )}}

Use the basic trigonometric identity:

cos (x) = \frac{1}{sec ( x )}

\frac{1}{\frac{1}{s e c ( t )} \cdot \frac{1}{c s c ( t )}}

Simplify

\frac{1}{\frac{1}{s e c ( t )} \cdot \frac{1}{c s c ( t )}}

Multiply

\frac{1}{sec ( t )} \cdot \frac{1}{csc ( t )} : \frac{1}{sec ( t ) csc ( t )}

\frac{1}{sec ( t )} \cdot \frac{1}{csc ( t )}

Multiply fractions:

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

= \frac{1 \cdot 1}{sec ( t ) csc ( t )}

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{sec ( t ) csc ( t )}

= \frac{1}{\frac{1}{s e c ( t ) c s c ( t )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{sec ( t ) csc ( t )}{1}

Apply rule

\frac{a}{1} = a

= sec (t) csc (t)

sec (t) csc (t)

sec (t) csc (t)

= csc (t) sec (t)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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