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Popular Trigonometria >

provar 2-2tan(x)cot(2x)=sec^2(x)

Solução

provar

2 - 2 tan (x) cot (2 x) = sec^{2} (x)

Solução

Verdadeiro

Passos da solução

2 - 2 tan (x) cot (2 x) = sec^{2} (x)

Manipular o lado direito

2 - 2 tan (x) cot (2 x)

Expresar com seno, cosseno

2 - 2 cot (2 x) tan (x)

Utilizar a Identidade Básica da Trigonometria:

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

= 2 - 2 \cdot \frac{cos ( 2 x )}{sin ( 2 x )} tan (x)

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

2 - 2 \cdot \frac{cos ( 2 x )}{sin ( 2 x )} \cdot \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( 2 x ) cos ( x ) - 2 cos ( 2 x ) sin ( x )}{sin ( 2 x ) cos ( x )}

2 - 2 \cdot \frac{cos ( 2 x )}{sin ( 2 x )} \cdot \frac{sin ( x )}{cos ( x )}

Multiplicar

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

2 \cdot \frac{cos ( 2 x )}{sin ( 2 x )} \cdot \frac{sin ( x )}{cos ( x )}

Multiplicar frações:

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

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

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

Converter para fração:

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

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

Já que os denominadores são iguais, combinar as frações:

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

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

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

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

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica do arco duplo:

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

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

Utilizar a identidade trigonométrica do arco duplo:

sin (2 x) = 2 sin (x) cos (x)

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

Simplificar

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

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

- 2 (2 cos^{2} (x) - 1) sin (x) + 2 cos (x) \cdot 2 sin (x) cos (x) = - 2 sin (x) (2 cos^{2} (x) - 1) + 4 cos^{2} (x) sin (x)

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

2 cos (x) \cdot 2 sin (x) cos (x) = 4 cos^{2} (x) sin (x)

2 cos (x) \cdot 2 sin (x) cos (x)

Multiplicar os números:

2 \cdot 2 = 4

= 4 cos (x) sin (x) cos (x)

Aplicar as propriedades dos expoentes:

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

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

= 4 sin (x) cos^{1 + 1} (x)

Somar:

1 + 1 = 2

= 4 sin (x) cos^{2} (x)

= - 2 sin (x) (2 cos^{2} (x) - 1) + 4 cos^{2} (x) sin (x)

= \frac{- 2 sin ( x ) ( 2 cos ^{2} ( x ) - 1 ) + 4 cos ^{2} ( x ) sin ( x )}{2 cos ( x ) sin ( x ) cos ( x )}

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

cos (x) \cdot 2 sin (x) cos (x)

Aplicar as propriedades dos expoentes:

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

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

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

Somar:

1 + 1 = 2

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

= \frac{- 2 sin ( x ) ( 2 cos ^{2} ( x ) - 1 ) + 4 cos ^{2} ( x ) sin ( x )}{2 cos ^{2} ( x ) sin ( x )}

Fatorar

- 2 (2 cos^{2} (x) - 1) sin (x) + 4 sin (x) cos^{2} (x) : 2 sin (x)

- 2 (2 cos^{2} (x) - 1) sin (x) + 4 sin (x) cos^{2} (x)

Reescrever como

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

Fatorar o termo comum

2 sin (x)

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

Expandir

2 cos^{2} (x) - (2 cos^{2} (x) - 1) : 1

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

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

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

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

Colocar os parênteses

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

Aplicar as regras dos sinais

- (- a) = a, - (a) = - a

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

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

Somar elementos similares:

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

= 1

= 2 \cdot 1 \cdot sin (x)

Simplificar

= 2 sin (x)

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

Cancelar

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

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

Dividir:

\frac{2}{2} = 1

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

Eliminar o fator comum:

sin (x)

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

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

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

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

Reeecreva usando identidades trigonométricas

Utilizar a Identidade Básica da Trigonometria:

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

\frac{1}{( \frac{1}{s e c ( x )} ) ^{2}}

Simplificar

\frac{1}{( \frac{1}{s e c ( x )} ) ^{2}}

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

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

Aplicar as propriedades dos expoentes:

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

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

Aplicar a regra

1^{a} = 1

1^{2} = 1

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

= \frac{1}{\frac{1}{s e c ^{2} ( x )}}

Aplicar as propriedades das frações:

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

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

Aplicar a regra

\frac{a}{1} = a

= sec^{2} (x)

sec^{2} (x)

sec^{2} (x)

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

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