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Popular >

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

Solución

verificar

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

Solución

Verdadero

Pasos de solución

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

Sea:

u = \frac{x}{2}

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

Verificar

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

Verdadero

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

Manipular el lado derecho

1 - tan^{2} (u)

Expresar con seno, coseno

1 - tan^{2} (u)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Aplicar las leyes de los exponentes:

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

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

Convertir a fracción:

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

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

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} ( u ) - sin ^{2} ( u )}{cos ^{2} ( u )}

Multiplicar:

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

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

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

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

Re-escribir usando identidades trigonométricas

\frac{cos ^{2} ( u ) - sin ^{2} ( u )}{cos ^{2} ( u )}

Utilizar la identidad pitagórica:

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

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

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

Expandir

cos^{2} (u) - (1 - cos^{2} (u)) : 2 cos^{2} (u) - 1

cos^{2} (u) - (1 - cos^{2} (u))

- (1 - cos^{2} (u)) : - 1 + cos^{2} (u)

- (1 - cos^{2} (u))

Poner los parentesis

= - (1) - (- cos^{2} (u))

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

= - 1 + cos^{2} (u)

= cos^{2} (u) - 1 + cos^{2} (u)

Simplificar

cos^{2} (u) - 1 + cos^{2} (u) : 2 cos^{2} (u) - 1

cos^{2} (u) - 1 + cos^{2} (u)

Agrupar términos semejantes

= cos^{2} (u) + cos^{2} (u) - 1

Sumar elementos similares:

cos^{2} (u) + cos^{2} (u) = 2 cos^{2} (u)

= 2 cos^{2} (u) - 1

= 2 cos^{2} (u) - 1

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

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

Manipular el lado izquierdo

\frac{2 cos ( 2 u )}{1 + cos ( 2 u )}

Re-escribir usando identidades trigonométricas

\frac{2 cos ( 2 u )}{1 + cos ( 2 u )}

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

1 + 2 cos^{2} (u) - 1 = 2 cos^{2} (u)

1 + 2 cos^{2} (u) - 1

Agrupar términos semejantes

= 2 cos^{2} (u) + 1 - 1

1 - 1 = 0

= 2 cos^{2} (u)

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

Dividir:

\frac{2}{2} = 1

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Por lo tanto

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Ejemplos populares

verificar (1+cos(θ))/(1+sec(θ))=cos(θ)verificar

\frac{1 + cos ( θ )}{1 + sec ( θ )} = cos (θ)

verificar tan^2(θ)-sec^2(θ)=-1 verificar

tan^{2} (θ) - sec^{2} (θ) = - 1

verificar cot(x)sin(x)-cos^2(x)sec(x)=0 verificar

cot (x) sin (x) - cos^{2} (x) sec (x) = 0

verificar (cos(3x))/(cos(x))=4cos^2(x)-3 verificar

\frac{cos ( 3 x )}{cos ( x )} = 4 cos^{2} (x) - 3

verificar (1-cos(2θ))/2 =sin^2(θ)verificar

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