English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometria >

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

Solução

provar

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

Solução

Verdadeiro

Passos da solução

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

Manipular o lado direito

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

Expresar com seno, cosseno

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

Utilizar a Identidade Básica da Trigonometria:

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)

Utilizar a Identidade Básica da Trigonometria:

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 )}

Utilizar a Identidade Básica da Trigonometria:

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 )}

Simplificar

- (\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 )}

Multiplicar frações:

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 )}

Multiplicar:

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}

Aplicar as propriedades dos expoentes:

(\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 )}

Aplicar as propriedades das frações:

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

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

Eliminar o fator comum:

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 )}

Multiplicar frações:

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 )}

Eliminar o fator comum:

cos (x)

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

Multiplicar:

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}

Aplicar as propriedades dos expoentes:

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

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

Aplicar a regra

1^{a} = 1

1^{2} = 1

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

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

Aplicar as propriedades das frações:

\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)

Aplicar as propriedades dos expoentes:

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

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

= sin^{2 + 1} (x)

Somar:

2 + 1 = 3

= sin^{3} (x)

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

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

Aplicar a regra

\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 )}

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica pitagórica:

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

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

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

Simplificar

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

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

Aplicar as propriedades dos expoentes:

\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 )}

Subtrair:

3 - 2 = 1

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

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

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

Reeecreva usando identidades trigonométricas

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

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

Aplicar as propriedades das frações:

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

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

Aplicar a regra

\frac{a}{1} = a

= csc (x)

csc (x)

csc (x)

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

Exemplos populares

provar csc(t)-sin(t)=cot(t)cos(t)

p ro v e csc (t) - sin (t) = cot (t) cos (t)

provar cos(60)=2cos^2(30)-1

p ro v e cos (6 0^{\circ}) = 2 cos^{2} (3 0^{\circ}) - 1

provar (sin(t)+cos(t))^2=1+sin(2t)

p ro v e (sin (t) + cos (t))^{2} = 1 + sin (2 t)

provar cos^2(x/2)=(sec(x)+1)/(2sec(x))

p ro v e cos^{2} (\frac{x}{2}) = \frac{sec ( x ) + 1}{2 sec ( x )}

provar sin(7x)+sin(x)=2cos(3x)sin(4x)

p ro v e sin (7 x) + sin (x) = 2 cos (3 x) sin (4 x)