Is cot(x)sec(x)csc^2(x)-cot^3(x)sec(x)=csc(x) ?

The answer to whether cot(x)sec(x)csc^2(x)-cot^3(x)sec(x)=csc(x) is True

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Popular Trigonometry >

prove cot(x)sec(x)csc^2(x)-cot^3(x)sec(x)=csc(x)

Solution

prove

cot (x) sec (x) csc^{2} (x) - cot^{3} (x) sec (x) = csc (x)

Solution

True

Solution steps

cot (x) sec (x) csc^{2} (x) - cot^{3} (x) sec (x) = csc (x)

Manipulating left side

cot (x) sec (x) csc^{2} (x) - cot^{3} (x) sec (x)

Express with sin, cos

- cot^{3} (x) sec (x) + cot (x) csc^{2} (x) sec (x)

Use the basic trigonometric identity:

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

= - (\frac{cos ( x )}{sin ( x )})^{3} sec (x) + \frac{cos ( x )}{sin ( x )} csc^{2} (x) sec (x)

Use the basic trigonometric identity:

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

= - (\frac{cos ( x )}{sin ( x )})^{3} \frac{1}{cos ( x )} + \frac{cos ( x )}{sin ( x )} csc^{2} (x) \frac{1}{cos ( x )}

Use the basic trigonometric identity:

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

= - (\frac{cos ( x )}{sin ( x )})^{3} \frac{1}{cos ( x )} + \frac{cos ( x )}{sin ( x )} (\frac{1}{sin ( x )})^{2} \frac{1}{cos ( x )}

Simplify

- (\frac{cos ( x )}{sin ( x )})^{3} \frac{1}{cos ( x )} + \frac{cos ( x )}{sin ( x )} (\frac{1}{sin ( x )})^{2} \frac{1}{cos ( x )} : \frac{- cos ^{2} ( x ) + 1}{sin ^{3} ( x )}

- (\frac{cos ( x )}{sin ( x )})^{3} \frac{1}{cos ( x )} + \frac{cos ( x )}{sin ( x )} (\frac{1}{sin ( x )})^{2} \frac{1}{cos ( x )}

(\frac{cos ( x )}{sin ( x )})^{3} \frac{1}{cos ( x )} = \frac{cos ^{2} ( x )}{sin ^{3} ( x )}

(\frac{cos ( x )}{sin ( x )})^{3} \frac{1}{cos ( x )}

Multiply fractions:

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

= \frac{1 \cdot ( \frac{c o s ( x )}{s i n ( x )} ) ^{3}}{cos ( x )}

Multiply:

1 \cdot (\frac{cos ( x )}{sin ( x )})^{3} = (\frac{cos ( x )}{sin ( x )})^{3}

= \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{3}}{cos ( x )}

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

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

Apply exponent rule:

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

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

= \frac{\frac{c o s ^{3} ( x )}{s i n ^{3} ( x )}}{cos ( x )}

Apply the fraction rule:

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

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

Cancel the common factor:

cos (x)

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

\frac{cos ( x )}{sin ( x )} (\frac{1}{sin ( x )})^{2} \frac{1}{cos ( x )} = \frac{1}{sin ^{3} ( x )}

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= \frac{1 \cdot ( \frac{1}{s i n ( x )} ) ^{2}}{sin ( x )}

Multiply:

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Apply the fraction rule:

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

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

sin^{2} (x) sin (x) = sin^{3} (x)

sin^{2} (x) sin (x)

Apply exponent rule:

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

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

= sin^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= sin^{3} (x)

= \frac{1}{sin ^{3} ( x )}

= - \frac{cos ^{2} ( x )}{sin ^{3} ( x )} + \frac{1}{sin ^{3} ( x )}

Apply rule

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

= \frac{- cos ^{2} ( x ) + 1}{sin ^{3} ( x )}

= \frac{- cos ^{2} ( x ) + 1}{sin ^{3} ( x )}

= \frac{1 - cos ^{2} ( x )}{sin ^{3} ( x )}

Rewrite using trig identities

\frac{1 - cos ^{2} ( x )}{sin ^{3} ( x )}

Use the Pythagorean identity:

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

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

= \frac{sin ^{2} ( x )}{sin ^{3} ( x )}

Simplify

\frac{sin ^{2} ( x )}{sin ^{3} ( x )} : \frac{1}{sin ( x )}

\frac{sin ^{2} ( x )}{sin ^{3} ( x )}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{sin ^{2} ( x )}{sin ^{3} ( x )} = \frac{1}{sin ^{3 - 2} ( x )}

= \frac{1}{sin ^{3 - 2} ( x )}

Subtract the numbers:

3 - 2 = 1

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{c s c ( x )}}

Simplify

\frac{1}{\frac{1}{c s c ( x )}}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= csc (x)

csc (x)

csc (x)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is cot(x)sec(x)csc^2(x)-cot^3(x)sec(x)=csc(x) ?
The answer to whether cot(x)sec(x)csc^2(x)-cot^3(x)sec(x)=csc(x) is True