English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

\frac{cos ( x )}{sec ( x ) - tan ( x )}

Expresar con seno, coseno

\frac{cos ( x )}{sec ( x ) - tan ( x )}

Utilizar la identidad trigonométrica básica:

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

= \frac{cos ( x )}{\frac{1}{c o s ( x )} - tan ( x )}

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Combinar las fracciones usando el mínimo común denominador:

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

Aplicar la regla

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

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

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

Aplicar las propiedades de las fracciones:

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Simplificar

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

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

Cancelar

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

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

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

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

Simplificar

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

Eliminar los terminos comunes:

sin (x) - 1

= sin (x) + 1

= sin (x) + 1

= sin (x) + 1

= sin (x) + 1

= 1 + sin (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

2 csc (2 x) = csc^{2} (x) tan (x)

\frac{sec ^{2} ( t )}{tan ( t )} = cot (t) + tan (t)

csc (\frac{π}{2} - x) = sec (x)

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