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

Solução

provar

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

Verdadeiro

Passos da solução

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

Manipular o lado direito

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

Reeecreva usando identidades trigonométricas

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

Multiplicar

\frac{1 + cos ( x )}{1 + cos ( x )}

por

Eq u a t i o n 1

= \frac{sec ( x )}{\frac{( 1 - c o s ( x ) ) ( 1 + c o s ( x ) )}{1 + c o s ( x )}}

Aplicar a regra da diferença de quadrados:

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

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

= \frac{sec ( x )}{\frac{1 - c o s ^{2} ( x )}{1 + c o s ( x )}}

Utilizar a identidade trigonométrica pitagórica:

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

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

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

Aplicar as propriedades das frações:

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

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

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

Expandir

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

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

Expandir

(1 + cos (x)) sec (x) : sec (x) + sec (x) cos (x)

(1 + cos (x)) sec (x)

= sec (x) (1 + cos (x))

Colocar os parênteses utilizando:

a (b + c) = ab + a c

a = sec (x), b = 1, c = cos (x)

= sec (x) \cdot 1 + sec (x) cos (x)

= 1 \cdot sec (x) + sec (x) cos (x)

Multiplicar:

1 \cdot sec (x) = sec (x)

= sec (x) + sec (x) cos (x)

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

Aplicar as propriedades das frações:

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

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

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

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

Reeecreva usando identidades trigonométricas

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

sec (x) cos (x) = 1

sec (x) cos (x)

Expresar com seno, cosseno

sec (x) cos (x)

Utilizar a Identidade Básica da Trigonometria:

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

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

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

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

Multiplicar frações:

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

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

Eliminar o fator comum:

cos (x)

= 1

= 1

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

Aplicar a regra

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

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

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

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

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

p ro v e tan (\frac{π}{4} - θ) = \frac{cos ( θ ) - sin ( θ )}{cos ( θ ) + sin ( θ )}

p ro v e cot (2 θ) = \frac{1}{2} sec (θ) csc (θ) - tan (θ)

p ro v e 8 cos^{2} (x) - 8 sin^{2} (x) = 8 - 16 sin^{2} (x)

p ro v e sec (x) - sin (x) \cdot tan (x) = cos (x)

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