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Popular >

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

Solución

verificar

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

Solución

Verdadero

Pasos de solución

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

Manipular el lado izquierdo

(sec (x) - tan (x)) sec (x)

Expresar con seno, coseno

(sec (x) - tan (x)) sec (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Quitar los parentesis:

(a) = a

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

Multiplicar fracciones:

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

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

(1 - sin (x)) \cdot 1 = 1 - sin (x)

(1 - sin (x)) \cdot 1

Multiplicar:

(1 - sin (x)) \cdot 1 = (1 - sin (x))

= (1 - sin (x))

Quitar los parentesis:

(a) = a

= 1 - sin (x)

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Cancelar

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

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

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

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

Simplificar

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

Eliminar los terminos comunes:

1 - sin (x)

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

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

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Ejemplos populares

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(sec (x) + tan (x))^{2} = 1

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tan (A) cot (A) = cos^{2} (A) + sin^{2} (A)

verificar tan(a)+cot(a)= 2/(sin(2a))verificar

tan (a) + cot (a) = \frac{2}{sin ( 2 a )}

verificar 1+sin^2(x)=2sin^2(x)+cos^2(x)verificar

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

verificar csc(x)= 1/(sec(x))verificar

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