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verificar (tan(a)sin(a))/(sec(a)-1)=1+cos(a)

Solución

verificar

\frac{tan ( a ) sin ( a )}{sec ( a ) - 1} = 1 + cos (a)

Solución

Verdadero

Pasos de solución

\frac{tan ( a ) sin ( a )}{sec ( a ) - 1} = 1 + cos (a)

Manipular el lado derecho

\frac{tan ( a ) sin ( a )}{sec ( a ) - 1}

Expresar con seno, coseno

\frac{sin ( a ) tan ( a )}{- 1 + sec ( a )}

Utilizar la identidad trigonométrica básica:

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

= \frac{sin ( a ) \frac{s i n ( a )}{c o s ( a )}}{- 1 + sec ( a )}

Utilizar la identidad trigonométrica básica:

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

= \frac{sin ( a ) \frac{s i n ( a )}{c o s ( a )}}{- 1 + \frac{1}{c o s ( a )}}

Simplificar

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

\frac{sin ( a ) \frac{s i n ( a )}{c o s ( a )}}{- 1 + \frac{1}{c o s ( a )}}

Simplificar

- 1 + \frac{1}{cos ( a )}

en una fracción:

\frac{- cos ( a ) + 1}{cos ( a )}

- 1 + \frac{1}{cos ( a )}

Convertir a fracción:

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

= - \frac{1 \cdot cos ( a )}{cos ( a )} + \frac{1}{cos ( a )}

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 ( a ) + 1}{cos ( a )}

Multiplicar:

1 \cdot cos (a) = cos (a)

= \frac{- cos ( a ) + 1}{cos ( a )}

= \frac{\frac{s i n ( a )}{c o s ( a )} sin ( a )}{\frac{- c o s ( a ) + 1}{c o s ( a )}}

Multiplicar

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

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

Multiplicar fracciones:

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

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

sin (a) sin (a) = sin^{2} (a)

sin (a) sin (a)

Aplicar las leyes de los exponentes:

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

sin (a) sin (a) = sin^{1 + 1} (a)

= sin^{1 + 1} (a)

Sumar:

1 + 1 = 2

= sin^{2} (a)

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

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

Dividir fracciones:

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

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

Eliminar los terminos comunes:

cos (a)

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

\frac{1 - cos ^{2} ( a )}{1 - cos ( a )}

Factorizar

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

1 - cos^{2} (a)

Factorizar el termino común

- 1

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

Factorizar

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

cos^{2} (a) - 1

Reescribir

1

como

1^{2}

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

= (cos (a) + 1) (cos (a) - 1)

= - (cos (a) + 1) (cos (a) - 1)

= - \frac{( cos ( a ) + 1 ) ( cos ( a ) - 1 )}{1 - cos ( a )}

Cancelar

- \frac{( cos ( a ) + 1 ) ( cos ( a ) - 1 )}{1 - cos ( a )} : cos (a) + 1

- \frac{( cos ( a ) + 1 ) ( cos ( a ) - 1 )}{1 - cos ( a )}

- cos (a) + 1 = - (cos (a) - 1)

= - \frac{( cos ( a ) + 1 ) ( cos ( a ) - 1 )}{- ( cos ( a ) - 1 )}

Simplificar

= \frac{( cos ( a ) + 1 ) ( cos ( a ) - 1 )}{cos ( a ) - 1}

Eliminar los terminos comunes:

cos (a) - 1

= cos (a) + 1

= cos (a) + 1

= cos (a) + 1

= cos (a) + 1

= 1 + cos (a)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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