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verificar (tan(x))/(1+sec(x))=-cot(x)+csc(x)

Solución

verificar

\frac{tan ( x )}{1 + sec ( x )} = - cot (x) + csc (x)

Solución

Verdadero

Pasos de solución

\frac{tan ( x )}{1 + sec ( x )} = - cot (x) + csc (x)

Manipular el lado derecho

\frac{tan ( x )}{1 + sec ( x )}

Expresar con seno, coseno

\frac{tan ( x )}{1 + sec ( x )}

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Aplicar las propiedades de las fracciones:

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

= \frac{sin ( x )}{cos ( x ) ( 1 + \frac{1}{c o s ( x )} )}

Simplificar

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

en una fracción:

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

Multiplicar:

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

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

= \frac{sin ( x )}{\frac{c o s ( x ) + 1}{c o s ( x )} cos ( x )}

Multiplicar

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos (x)

= cos (x) + 1

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

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

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

Manipular el lado izquierdo

- cot (x) + csc (x)

Expresar con seno, coseno

- cot (x) + csc (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Aplicar la regla

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

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

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

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

Re-escribir usando identidades trigonométricas

Multiplicar

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

por

Eq u a t i o n 1

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

Expandir

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

(1 + cos (x)) (1 - cos (x))

Aplicar la siguiente regla para binomios al cuadrado:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = cos (x)

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

= 1 - cos^{2} (x)

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

Utilizar la identidad pitagórica:

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

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

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

Eliminar los terminos comunes:

sin (x)

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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