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

Solución

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

Solución

Sin soluci \overset{o}{ˊ} n para x \in R

Pasos de solución

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

Restar

tan^{2} (x)

de ambos lados

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

Expresar con seno, coseno

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

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

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

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

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

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

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

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

Combinar las fracciones usando el mínimo común denominador:

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

Aplicar la regla

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

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

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

Convertir a fracción:

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

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

Multiplicar:

1 \cdot cos^{2} (x) = cos^{2} (x)

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

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

\frac{1 + cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

1 + cos^{2} (x) - sin^{2} (x) = 0

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

= 1 + cos (2 x)

1 + cos (2 x) = 0

Mover

1

al lado derecho

1 + cos (2 x) = 0

Restar

1

de ambos lados

1 + cos (2 x) - 1 = 0 - 1

Simplificar

cos (2 x) = - 1

cos (2 x) = - 1

Soluciones generales para

cos (2 x) = - 1

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

2 x = π + 2 πn

2 x = π + 2 πn

Resolver

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Dividir ambos lados entre

2

2 x = π + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

Siendo que la ecuación esta indefinida para:

\frac{π}{2} + πn

Sin soluci \overset{o}{ˊ} n para x \in R

Ejemplos populares

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cos (t) = \frac{1}{2}

3cos^2(x)+8sin(x)=7

3 cos^{2} (x) + 8 sin (x) = 7

csc(x+10)=3

csc (x + 1 0^{\circ}) = 3

6tan(x)=18cot(x)

6 tan (x) = 18 cot (x)

9cos(x)=0

9 cos (x) = 0