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Popular >

verificar (cos(x)cot(x))/(1-sin(x))=csc(x)+1

Solución

verificar

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} = csc (x) + 1

Solución

Verdadero

Pasos de solución

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} = csc (x) + 1

Manipular el lado derecho

\frac{cos ( x ) cot ( x )}{1 - sin ( x )}

Expresar con seno, coseno

\frac{cos ( x ) cot ( x )}{1 - sin ( x )}

Utilizar la identidad trigonométrica básica:

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

= \frac{cos ( x ) \frac{c o s ( x )}{s i n ( x )}}{1 - sin ( x )}

Simplificar

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

\frac{cos ( x ) \frac{c o s ( x )}{s i n ( x )}}{1 - sin ( x )}

Multiplicar

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

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

Multiplicar fracciones:

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

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

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

cos (x) cos (x)

Aplicar las leyes de los exponentes:

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

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

= cos^{1 + 1} (x)

Sumar:

1 + 1 = 2

= cos^{2} (x)

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

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

Aplicar las propiedades de las fracciones:

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Factorizar

1 - sin^{2} (x) : - (sin (x) + 1) (sin (x) - 1)

1 - sin^{2} (x)

Factorizar el termino común

- 1

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

Factorizar

sin^{2} (x) - 1 : (sin (x) + 1) (sin (x) - 1)

sin^{2} (x) - 1

Reescribir

1

como

1^{2}

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

= (sin (x) + 1) (sin (x) - 1)

= - (sin (x) + 1) (sin (x) - 1)

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

Factorizar

1 - sin (x) : - (sin (x) - 1)

1 - sin (x)

Factorizar el termino común

- 1

= - (sin (x) - 1)

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

Eliminar los terminos comunes:

sin (x) - 1

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

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

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

Re-escribir usando identidades trigonométricas

Utilizar la identidad trigonométrica básica:

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

\frac{\frac{1}{c s c ( x )} + 1}{\frac{1}{c s c ( x )}}

Simplificar

\frac{\frac{1}{c s c ( x )} + 1}{\frac{1}{c s c ( x )}}

Aplicar las propiedades de las fracciones:

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

= \frac{( \frac{1}{c s c ( x )} + 1 ) csc ( x )}{1}

Simplificar

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

en una fracción:

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

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

Convertir a fracción:

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

= \frac{1}{csc ( x )} + \frac{1 \cdot csc ( x )}{csc ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{1 + 1 \cdot csc ( x )}{csc ( x )}

Multiplicar:

1 \cdot csc (x) = csc (x)

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

= \frac{\frac{c s c ( x ) + 1}{c s c ( x )} csc ( x )}{1}

Aplicar las propiedades de las fracciones:

\frac{a}{1} = a

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

csc (x)

= 1 + csc (x)

1 + csc (x)

1 + csc (x)

= csc (x) + 1

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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