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Popular >

verificar (tan(θ)+cot(θ))/(tan(θ))=csc^2(θ)

Solución

verificar

\frac{tan ( θ ) + cot ( θ )}{tan ( θ )} = csc^{2} (θ)

Solución

Verdadero

Pasos de solución

\frac{tan ( θ ) + cot ( θ )}{tan ( θ )} = csc^{2} (θ)

Manipular el lado derecho

\frac{tan ( θ ) + cot ( θ )}{tan ( θ )}

Expresar con seno, coseno

\frac{cot ( θ ) + tan ( θ )}{tan ( θ )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{c o s ( θ )}{s i n ( θ )} + tan ( θ )}{tan ( θ )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{c o s ( θ )}{s i n ( θ )} + \frac{s i n ( θ )}{c o s ( θ )}}{\frac{s i n ( θ )}{c o s ( θ )}}

Simplificar

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

\frac{\frac{c o s ( θ )}{s i n ( θ )} + \frac{s i n ( θ )}{c o s ( θ )}}{\frac{s i n ( θ )}{c o s ( θ )}}

Aplicar las propiedades de las fracciones:

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

= \frac{( \frac{c o s ( θ )}{s i n ( θ )} + \frac{s i n ( θ )}{c o s ( θ )} ) cos ( θ )}{sin ( θ )}

Simplificar

\frac{cos ( θ )}{sin ( θ )} + \frac{sin ( θ )}{cos ( θ )}

en una fracción:

\frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ( θ ) cos ( θ )}

\frac{cos ( θ )}{sin ( θ )} + \frac{sin ( θ )}{cos ( θ )}

Mínimo común múltiplo de

sin (θ), cos (θ) : sin (θ) cos (θ)

sin (θ), cos (θ)

Mínimo común múltiplo (MCM)

Calcular una expresión que este compuesta de factores que aparezcan tanto en

sin (θ)

cos (θ)

= sin (θ) cos (θ)

Reescribir las fracciones basandose en el mínimo común denominador

Multiplicar cada numerador por la misma cantidad necesaria para multiplicar el denominador correspondiente y convertirlo en el mínimo común denominador

Para

\frac{cos ( θ )}{sin ( θ )} :

multiplicar el denominador y el numerador por

cos (θ)

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

Para

\frac{sin ( θ )}{cos ( θ )} :

multiplicar el denominador y el numerador por

sin (θ)

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

= \frac{\frac{c o s ^{2} ( θ ) + s i n ^{2} ( θ )}{s i n ( θ ) c o s ( θ )} cos ( θ )}{sin ( θ )}

Multiplicar

\frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ( θ ) cos ( θ )} cos (θ) : \frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ( θ )}

\frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ( θ ) cos ( θ )} cos (θ)

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos (θ)

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

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

Aplicar las propiedades de las fracciones:

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

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

sin (θ) sin (θ) = sin^{2} (θ)

sin (θ) sin (θ)

Aplicar las leyes de los exponentes:

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

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= sin^{1 + 1} (θ)

Sumar:

1 + 1 = 2

= sin^{2} (θ)

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

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

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

Re-escribir usando identidades trigonométricas

\frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ^{2} ( θ )}

Utilizar la identidad pitagórica:

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

= \frac{1}{sin ^{2} ( θ )}

= \frac{1}{sin ^{2} ( θ )}

Re-escribir usando identidades trigonométricas

Utilizar la identidad trigonométrica básica:

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

\frac{1}{( \frac{1}{c s c ( θ )} ) ^{2}}

Simplificar

\frac{1}{( \frac{1}{c s c ( θ )} ) ^{2}}

(\frac{1}{csc ( θ )})^{2} = \frac{1}{csc ^{2} ( θ )}

(\frac{1}{csc ( θ )})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{1 ^{2}}{csc ^{2} ( θ )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( θ )}

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

Aplicar las propiedades de las fracciones:

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

= \frac{csc ^{2} ( θ )}{1}

Aplicar la regla

\frac{a}{1} = a

= csc^{2} (θ)

csc^{2} (θ)

csc^{2} (θ)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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