verificar tan^2(x)-(sec(x)+1)(sec(x)-1)=0

Solución

verificar

tan^{2} (x) - (sec (x) + 1) (sec (x) - 1) = 0

Verdadero

Pasos de solución

tan^{2} (x) - (sec (x) + 1) (sec (x) - 1) = 0

Manipular el lado derecho

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

Re-escribir usando identidades trigonométricas

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

Expandir

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

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

Aplicar la siguiente regla para binomios al cuadrado:

(a + b) (a - b) = a^{2} - b^{2}

a = sec (x), b = 1

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

= sec^{2} (x) - 1

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

Utilizar la identidad pitagórica:

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

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

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

Sumar elementos similares:

tan^{2} (x) - tan^{2} (x) = 0

= 0

= 0

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero