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provar (sin(x))/(cot^2(x))=(tan^2(x))/(csc(x))

Solução

provar

\frac{sin ( x )}{cot ^{2} ( x )} = \frac{tan ^{2} ( x )}{csc ( x )}

Verdadeiro

Passos da solução

\frac{sin ( x )}{cot ^{2} ( x )} = \frac{tan ^{2} ( x )}{csc ( x )}

Manipular o lado direito

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

Expresar com seno, cosseno

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

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

\frac{sin ( x )}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}} : \frac{sin ^{3} ( x )}{cos ^{2} ( x )}

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

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

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

Aplicar as propriedades dos expoentes:

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

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

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

Aplicar as propriedades das frações:

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

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

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

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

Aplicar as propriedades dos expoentes:

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

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

= sin^{1 + 2} (x)

Somar:

1 + 2 = 3

= sin^{3} (x)

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

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

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

Manipular o lado esquerdo

\frac{tan ^{2} ( x )}{csc ( x )}

Expresar com seno, cosseno

\frac{tan ^{2} ( x )}{csc ( x )}

Utilizar a Identidade Básica da Trigonometria:

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

= \frac{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{csc ( x )}

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

\frac{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{\frac{1}{s i n ( x )}} : \frac{sin ^{3} ( x )}{cos ^{2} ( x )}

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

Aplicar as propriedades das frações:

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

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

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

Multiplicar

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

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

Multiplicar frações:

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

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

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

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

Aplicar as propriedades dos expoentes:

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

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

= sin^{2 + 1} (x)

Somar:

2 + 1 = 3

= sin^{3} (x)

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

= \frac{\frac{s i n ^{3} ( x )}{c o s ^{2} ( x )}}{1}

Aplicar as propriedades das frações:

\frac{a}{1} = a

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

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

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

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

p ro v e sin (x) \cdot (1 + cot (x)) = sin (x) + cos (x)

p ro v e sin (\frac{5 π}{4}) = cos (\frac{5 π}{4})

p ro v e tan (13 5^{\circ} + θ) = \frac{- 1 + tan ( θ )}{1 + tan ( θ )}

p ro v e \frac{csc ( \frac{π}{2} - x )}{- tan ( - x )} = csc (x)

p ro v e tan (12 0^{\circ}) = tan (18 0^{\circ} - 6 0^{\circ})