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verificar csc^4(θ)-cot^4(θ)=2csc^2(θ)-1

Solución

verificar

csc^{4} (θ) - cot^{4} (θ) = 2 csc^{2} (θ) - 1

Verdadero

Pasos de solución

csc^{4} (θ) - cot^{4} (θ) = 2 csc^{2} (θ) - 1

Manipular el lado derecho

csc^{4} (θ) - cot^{4} (θ)

Factorizar

- cot^{4} (θ) + csc^{4} (θ) : (csc^{2} (θ) + cot^{2} (θ)) (csc (θ) + cot (θ)) (csc (θ) - cot (θ))

- cot^{4} (θ) + csc^{4} (θ)

Reescribir

csc^{4} (θ) - cot^{4} (θ)

como

(csc^{2} (θ))^{2} - (cot^{2} (θ))^{2}

csc^{4} (θ) - cot^{4} (θ)

Aplicar las leyes de los exponentes:

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

cot^{4} (θ) = (cot^{2} (θ))^{2}

= csc^{4} (θ) - (cot^{2} (θ))^{2}

Aplicar las leyes de los exponentes:

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

csc^{4} (θ) = (csc^{2} (θ))^{2}

= (csc^{2} (θ))^{2} - (cot^{2} (θ))^{2}

= (csc^{2} (θ))^{2} - (cot^{2} (θ))^{2}

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

(csc^{2} (θ))^{2} - (cot^{2} (θ))^{2} = (csc^{2} (θ) + cot^{2} (θ)) (csc^{2} (θ) - cot^{2} (θ))

= (csc^{2} (θ) + cot^{2} (θ)) (csc^{2} (θ) - cot^{2} (θ))

Factorizar

csc^{2} (θ) - cot^{2} (θ) : (csc (θ) + cot (θ)) (csc (θ) - cot (θ))

csc^{2} (θ) - cot^{2} (θ)

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

csc^{2} (θ) - cot^{2} (θ) = (csc (θ) + cot (θ)) (csc (θ) - cot (θ))

= (csc (θ) + cot (θ)) (csc (θ) - cot (θ))

= (csc^{2} (θ) + cot^{2} (θ)) (csc (θ) + cot (θ)) (csc (θ) - cot (θ))

= (- cot (θ) + csc (θ)) (cot (θ) + csc (θ)) (cot^{2} (θ) + csc^{2} (θ))

Re-escribir usando identidades trigonométricas

(- cot (θ) + csc (θ)) (cot (θ) + csc (θ)) (cot^{2} (θ) + csc^{2} (θ))

Utilizar la identidad pitagórica:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= (- cot (θ) + csc (θ)) (cot (θ) + csc (θ)) (csc^{2} (θ) - 1 + csc^{2} (θ))

Simplificar

csc^{2} (θ) - 1 + csc^{2} (θ) : 2 csc^{2} (θ) - 1

csc^{2} (θ) - 1 + csc^{2} (θ)

Agrupar términos semejantes

= csc^{2} (θ) + csc^{2} (θ) - 1

Sumar elementos similares:

csc^{2} (θ) + csc^{2} (θ) = 2 csc^{2} (θ)

= 2 csc^{2} (θ) - 1

= (csc (θ) - cot (θ)) (cot (θ) + csc (θ)) (2 csc^{2} (θ) - 1)

Expandir

(csc (θ) + cot (θ)) (csc (θ) - cot (θ)) : csc^{2} (θ) - cot^{2} (θ)

(csc (θ) + cot (θ)) (csc (θ) - cot (θ))

Aplicar la siguiente regla para binomios al cuadrado:

(a + b) (a - b) = a^{2} - b^{2}

a = csc (θ), b = cot (θ)

= csc^{2} (θ) - cot^{2} (θ)

= (- 1 + 2 csc^{2} (θ)) (csc^{2} (θ) - cot^{2} (θ))

Utilizar la identidad pitagórica:

csc^{2} (θ) = 1 + cot^{2} (θ)

csc^{2} (θ) - cot^{2} (θ) = 1

= (- 1 + 2 csc^{2} (θ)) \cdot 1

Simplificar

= - 1 + 2 csc^{2} (θ)

= - 1 + 2 csc^{2} (θ)

= 2 csc^{2} (θ) - 1

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero