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tan^2(x)+1=a^2

Solución

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

Solución

x = arctan (a^{2} - 1) + πn, x = arctan (- a^{2} - 1) + πn

Pasos de solución

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

Usando el método de sustitución

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

Sea:

tan (x) = u

u^{2} + 1 = a^{2}

u^{2} + 1 = a^{2} : u = a^{2} - 1, u = - a^{2} - 1

u^{2} + 1 = a^{2}

Mover

1

al lado derecho

u^{2} + 1 = a^{2}

Restar

1

de ambos lados

u^{2} + 1 - 1 = a^{2} - 1

Simplificar

u^{2} = a^{2} - 1

u^{2} = a^{2} - 1

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = a^{2} - 1, u = - a^{2} - 1

Sustituir en la ecuación

u = tan (x)

tan (x) = a^{2} - 1, tan (x) = - a^{2} - 1

tan (x) = a^{2} - 1, tan (x) = - a^{2} - 1

tan (x) = a^{2} - 1 : x = arctan (a^{2} - 1) + πn

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

Aplicar propiedades trigonométricas inversas

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

Soluciones generales para

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

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (a^{2} - 1) + πn

x = arctan (a^{2} - 1) + πn

tan (x) = - a^{2} - 1 : x = arctan (- a^{2} - 1) + πn

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

Aplicar propiedades trigonométricas inversas

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

Soluciones generales para

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

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (- a^{2} - 1) + πn

x = arctan (- a^{2} - 1) + πn

Combinar toda las soluciones

x = arctan (a^{2} - 1) + πn, x = arctan (- a^{2} - 1) + πn

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