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provar cot(x)-cot(x)cos^2(x)=sin(x)cos(x)

Solução

provar

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

Verdadeiro

Passos da solução

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

Manipular o lado direito

cot (x) - cot (x) cos^{2} (x)

Expresar com seno, cosseno

cot (x) - cos^{2} (x) cot (x)

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

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

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

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

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

Multiplicar frações:

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

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

cos (x) cos^{2} (x) = cos^{3} (x)

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

Aplicar as propriedades dos expoentes:

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

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

= cos^{1 + 2} (x)

Somar:

1 + 2 = 3

= cos^{3} (x)

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

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

Aplicar a regra

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

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

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

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

Fatorar

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

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

Fatorar

cos (x) - cos^{3} (x) : cos (x) (1 - cos^{2} (x))

cos (x) - cos^{3} (x)

Aplicar as propriedades dos expoentes:

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

cos^{3} (x) = cos (x) cos^{2} (x)

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

Fatorar o termo comum

cos (x)

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

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

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

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica pitagórica:

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

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

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

Eliminar o fator comum:

sin (x)

= sin (x) cos (x)

= sin (x) cos (x)

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

p ro v e \frac{1}{sec ( θ )} = arcsec (θ)

p ro v e \frac{( cot ( x ) )}{( csc ( x ) )} = cos (x)

p ro v e \frac{( sec ^{2} ( t ) )}{sec ^{2} ( t ) - 1} = csc^{2} (t)

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

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