English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

verificar cot^2(θ)=cos^2(θ)+(cot(θ)cos(θ))^2

Solución

verificar

cot^{2} (θ) = cos^{2} (θ) + (cot (θ) cos (θ))^{2}

Solución

Verdadero

Pasos de solución

cot^{2} (θ) = cos^{2} (θ) + (cot (θ) cos (θ))^{2}

Manipular el lado izquierdo

cos^{2} (θ) + (cot (θ) cos (θ))^{2}

Expresar con seno, coseno

(cos (θ) cot (θ))^{2} + cos^{2} (θ)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

(cos (θ) \frac{cos ( θ )}{sin ( θ )})^{2} + cos^{2} (θ) : \frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

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

(cos (θ) \frac{cos ( θ )}{sin ( θ )})^{2} = \frac{cos ^{4} ( θ )}{sin ^{2} ( θ )}

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

Multiplicar

cos (θ) \frac{cos ( θ )}{sin ( θ )} : \frac{cos ^{2} ( θ )}{sin ( θ )}

cos (θ) \frac{cos ( θ )}{sin ( θ )}

Multiplicar fracciones:

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

= \frac{cos ( θ ) cos ( θ )}{sin ( θ )}

cos (θ) cos (θ) = cos^{2} (θ)

cos (θ) cos (θ)

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

cos (θ) cos (θ) = cos^{1 + 1} (θ)

= cos^{1 + 1} (θ)

Sumar:

1 + 1 = 2

= cos^{2} (θ)

= \frac{cos ^{2} ( θ )}{sin ( θ )}

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

Aplicar las leyes de los exponentes:

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

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

(cos^{2} (θ))^{2} : cos^{4} (θ)

Aplicar las leyes de los exponentes:

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

= cos^{2 \cdot 2} (θ)

Multiplicar los numeros:

2 \cdot 2 = 4

= cos^{4} (θ)

= \frac{cos ^{4} ( θ )}{sin ^{2} ( θ )}

= \frac{cos ^{4} ( θ )}{sin ^{2} ( θ )} + cos^{2} (θ)

Convertir a fracción:

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

= \frac{cos ^{4} ( θ )}{sin ^{2} ( θ )} + \frac{cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

= \frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

= \frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

Factorizar

\frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )} : \frac{cos ^{2} ( θ ) ( cos ^{2} ( θ ) + sin ^{2} ( θ ) )}{sin ^{2} ( θ )}

\frac{cos ^{4} ( θ ) + cos ^{2} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ )}

Factorizar

cos^{4} (θ) + cos^{2} (θ) sin^{2} (θ) : cos^{2} (θ) (cos^{2} (θ) + sin^{2} (θ))

cos^{4} (θ) + cos^{2} (θ) sin^{2} (θ)

Aplicar las leyes de los exponentes:

a^{b + c} = a^{b} a^{c}

cos^{4} (θ) = cos^{2} (θ) cos^{2} (θ)

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

Factorizar el termino común

cos^{2} (θ)

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

Simplificar

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

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

Re-escribir usando identidades trigonométricas

= \frac{cos ( θ )}{sin ( θ )} \cdot \frac{cos ( θ )}{sin ( θ )}

Utilizar la identidad trigonométrica básica:

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

\frac{cos ( θ ) cot ( θ )}{sin ( θ )}

= cot (θ) \frac{cos ( θ )}{sin ( θ )}

Utilizar la identidad trigonométrica básica:

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

cot (θ) cot (θ)

Simplificar

cot (θ) cot (θ) : cot^{2} (θ)

cot (θ) cot (θ)

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

cot (θ) cot (θ) = cot^{1 + 1} (θ)

= cot^{1 + 1} (θ)

Sumar:

1 + 1 = 2

= cot^{2} (θ)

cot^{2} (θ)

cot^{2} (θ)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Ejemplos populares

verificar csc^2(θ)-1=cot^2(θ)verificar

csc^{2} (θ) - 1 = cot^{2} (θ)

verificar tan(u)+cot(u)=sec(u)csc(u)verificar

tan (u) + cot (u) = sec (u) csc (u)

verificar ((cot(a)))/((csc(a)))=cos(a)verificar

\frac{( cot ( a ) )}{( csc ( a ) )} = cos (a)

verificar tan(θ)=sqrt(tan(θ)+2)verificar

tan (θ) = tan (θ) + 2

verificar tan(2u)=(2tan(u))/(1-tan^2(u))verificar

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