English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometria >

provar (cos(x)cot(x))/(1-sin(x))-1=csc(x)

Solução

provar

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} - 1 = csc (x)

Solução

Verdadeiro

Passos da solução

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} - 1 = csc (x)

Manipular o lado direito

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} - 1

Expresar com seno, cosseno

- 1 + \frac{cos ( x ) cot ( x )}{1 - sin ( x )}

Utilizar a Identidade Básica da Trigonometria:

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

= - 1 + \frac{cos ( x ) \frac{c o s ( x )}{s i n ( x )}}{1 - sin ( x )}

Simplificar

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

- 1 + \frac{cos ( x ) \frac{c o s ( x )}{s i n ( x )}}{1 - sin ( x )}

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

\frac{cos ( x ) \frac{c o s ( x )}{s i n ( x )}}{1 - sin ( x )}

Multiplicar

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

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

Multiplicar frações:

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

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

cos (x) cos (x) = cos^{2} (x)

cos (x) cos (x)

Aplicar as propriedades dos expoentes:

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

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

= cos^{1 + 1} (x)

Somar:

1 + 1 = 2

= cos^{2} (x)

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

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

Aplicar as propriedades das frações:

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

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

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

Converter para fração:

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

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

Já que os denominadores são iguais, combinar as frações:

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

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

Multiplicar:

1 \cdot sin (x) = sin (x)

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

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

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

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica pitagórica:

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

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

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

Simplificar

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

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

Expandir

1 - sin^{2} (x) - (1 - sin (x)) sin (x) : - sin (x) + 1

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

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

Expandir

- sin (x) (1 - sin (x)) : - sin (x) + sin^{2} (x)

- sin (x) (1 - sin (x))

Colocar os parênteses utilizando:

a (b - c) = ab - a c

a = - sin (x), b = 1, c = sin (x)

= - sin (x) \cdot 1 - (- sin (x)) sin (x)

Aplicar as regras dos sinais

- (- a) = a

= - 1 \cdot sin (x) + sin (x) sin (x)

Simplificar

- 1 \cdot sin (x) + sin (x) sin (x) : - sin (x) + sin^{2} (x)

- 1 \cdot sin (x) + sin (x) sin (x)

1 \cdot sin (x) = sin (x)

1 \cdot sin (x)

Multiplicar:

1 \cdot sin (x) = sin (x)

= sin (x)

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

sin (x) sin (x)

Aplicar as propriedades dos expoentes:

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

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

= sin^{1 + 1} (x)

Somar:

1 + 1 = 2

= sin^{2} (x)

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

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

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

Simplificar

1 - sin^{2} (x) - sin (x) + sin^{2} (x) : - sin (x) + 1

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

Agrupar termos semelhantes

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

Somar elementos similares:

- sin^{2} (x) + sin^{2} (x) = 0

= - sin (x) + 1

= - sin (x) + 1

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

Eliminar o fator comum:

- sin (x) + 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 (1+tan(x))^2=sec^2(x)+2tan(x)

p ro v e (1 + tan (x))^{2} = sec^{2} (x) + 2 tan (x)

provar (sec^2(x))/(2tan(x))=csc(2x)

p ro v e \frac{sec ^{2} ( x )}{2 tan ( x )} = csc (2 x)

provar (sec^2(x)-1)csc(x)=tan(x)sec(x)

p ro v e (sec^{2} (x) - 1) csc (x) = tan (x) sec (x)

provar cos(x-pi/2)=sin(x)

p ro v e cos (x - \frac{π}{2}) = sin (x)

provar (1+tan^2(x))cot(x)=sec(x)csc(x)

p ro v e (1 + tan^{2} (x)) cot (x) = sec (x) csc (x)