English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

verificar cos^2(x)(2+tan^2(x))=2-sin^2(x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

cos^{2} (x) (2 + tan^{2} (x))

Expresar con seno, coseno

(2 + tan^{2} (x)) cos^{2} (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Aplicar las leyes de los exponentes:

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

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

Simplificar

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

en una fracción:

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

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

Convertir a fracción:

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

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos^{2} (x)

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

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

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

Manipular el lado izquierdo

2 - sin^{2} (x)

Re-escribir usando identidades trigonométricas

2 - sin^{2} (x)

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

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

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

Poner los parentesis

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

Aplicar las reglas de los signos

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

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

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

Restar:

2 - 1 = 1

= cos^{2} (x) + 1

= cos^{2} (x) + 1

Utilizar la identidad pitagórica:

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

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

Sumar elementos similares:

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero