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verificar 1+1/(cos(x))=(tan^2(x))/(sec(x)-1)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado izquierdo

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Factorizar

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

sec^{2} (x) - 1

Reescribir

1

como

1^{2}

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

= (sec (x) + 1) (sec (x) - 1)

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

Eliminar los terminos comunes:

sec (x) - 1

= sec (x) + 1

= sec (x) + 1

= sec (x) + 1

Expresar con seno, coseno

1 + sec (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Expandir

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

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

Aplicar las propiedades de las fracciones:

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

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

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

Aplicar la regla

\frac{a}{a} = 1

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

tan (θ) = 2

sec (x) = sec (- x)

csc^{2} (θ) - 1 = cot^{2} (θ)

\frac{( cot ( a ) )}{( csc ( a ) )} = cos (a)