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provar (tan^2(x)+1)(cos^2(x)-1)=-tan^2(x)

Solução

provar

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

Verdadeiro

Passos da solução

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

Manipular o lado direito

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

Expandir

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

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

Aplique o método FOIL:

(a + b) (c + d) = a c + a d + b c + b d

a = - 1, b = cos^{2} (x), c = 1, d = tan^{2} (x)

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

Aplicar as regras dos sinais

+ (- a) = - a

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

Simplificar

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

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

Multiplicar os números:

1 \cdot 1 = 1

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

Multiplicar:

1 \cdot tan^{2} (x) = tan^{2} (x)

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

Multiplicar:

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

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

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

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

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica pitagórica:

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

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

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

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

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

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

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

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

Aplicar as propriedades dos expoentes:

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

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

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

Multiplicar frações:

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

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

Eliminar o fator comum:

cos^{2} (x)

= sin^{2} (x)

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

Somar elementos similares:

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

= - tan^{2} (x)

= - tan^{2} (x)

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

p ro v e csc (θ) cos (θ) = cot (θ)

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

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

p ro v e cot (x) - tan (x) = 2 cot (2 x)

p ro v e sin^{4} (x) - cos^{4} (x) = 2 sin^{2} (x) - 1