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Popular >

verificar tan(x/2)=(tan(x))/((sec(x)+1))

Solución

verificar

tan (\frac{x}{2}) = \frac{tan ( x )}{( sec ( x ) + 1 )}

Solución

Verdadero

Pasos de solución

tan (\frac{x}{2}) = \frac{tan ( x )}{sec ( x ) + 1}

Sea:

u = \frac{x}{2}

tan (u) = \frac{tan ( 2 u )}{sec ( 2 u ) + 1}

Verificar

tan (u) = \frac{tan ( 2 u )}{sec ( 2 u ) + 1} :

Verdadero

tan (u) = \frac{tan ( 2 u )}{sec ( 2 u ) + 1}

Manipular el lado izquierdo

\frac{tan ( 2 u )}{sec ( 2 u ) + 1}

Expresar con seno, coseno

\frac{tan ( 2 u )}{1 + sec ( 2 u )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{s i n ( 2 u )}{c o s ( 2 u )}}{1 + sec ( 2 u )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{s i n ( 2 u )}{c o s ( 2 u )}}{1 + \frac{1}{c o s ( 2 u )}}

Simplificar

\frac{\frac{s i n ( 2 u )}{c o s ( 2 u )}}{1 + \frac{1}{c o s ( 2 u )}} : \frac{sin ( 2 u )}{cos ( 2 u ) + 1}

\frac{\frac{s i n ( 2 u )}{c o s ( 2 u )}}{1 + \frac{1}{c o s ( 2 u )}}

Aplicar las propiedades de las fracciones:

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

= \frac{sin ( 2 u )}{cos ( 2 u ) ( 1 + \frac{1}{c o s ( 2 u )} )}

Simplificar

1 + \frac{1}{cos ( 2 u )}

en una fracción:

\frac{cos ( 2 u ) + 1}{cos ( 2 u )}

1 + \frac{1}{cos ( 2 u )}

Convertir a fracción:

1 = \frac{1 cos ( 2 u )}{cos ( 2 u )}

= \frac{1 \cdot cos ( 2 u )}{cos ( 2 u )} + \frac{1}{cos ( 2 u )}

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 ( 2 u ) + 1}{cos ( 2 u )}

Multiplicar:

1 \cdot cos (2 u) = cos (2 u)

= \frac{cos ( 2 u ) + 1}{cos ( 2 u )}

= \frac{sin ( 2 u )}{\frac{c o s ( 2 u ) + 1}{c o s ( 2 u )} cos ( 2 u )}

Multiplicar

cos (2 u) \frac{cos ( 2 u ) + 1}{cos ( 2 u )} : cos (2 u) + 1

cos (2 u) \frac{cos ( 2 u ) + 1}{cos ( 2 u )}

Multiplicar fracciones:

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

= \frac{( cos ( 2 u ) + 1 ) cos ( 2 u )}{cos ( 2 u )}

Eliminar los terminos comunes:

cos (2 u)

= cos (2 u) + 1

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

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

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

Re-escribir usando identidades trigonométricas

\frac{sin ( 2 u )}{1 + cos ( 2 u )}

Utilizar la identidad trigonométrica del ángulo doble:

sin (2 x) = 2 sin (x) cos (x)

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

1 + 2 cos^{2} (u) - 1 = 2 cos^{2} (u)

1 + 2 cos^{2} (u) - 1

Agrupar términos semejantes

= 2 cos^{2} (u) + 1 - 1

1 - 1 = 0

= 2 cos^{2} (u)

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

Dividir:

\frac{2}{2} = 1

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

Eliminar los terminos comunes:

cos (u)

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

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

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

Utilizar la identidad trigonométrica básica:

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

= tan (u)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Por lo tanto

tan (\frac{x}{2}) = \frac{tan ( x )}{sec ( x ) + 1}

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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