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Popular >

verificar cos(x)-csc(x)cot(x)=-cos(x)cot^2(x)

Solución

verificar

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

Solución

Verdadero

Pasos de solución

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

Manipular el lado derecho

cos (x) - csc (x) cot (x)

Expresar con seno, coseno

cos (x) - cot (x) csc (x)

Utilizar la identidad trigonométrica básica:

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

= cos (x) - \frac{cos ( x )}{sin ( x )} csc (x)

Utilizar la identidad trigonométrica básica:

csc (x) = \frac{1}{sin ( x )}

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

Simplificar

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

cos (x) - \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

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

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

Multiplicar fracciones:

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

= \frac{cos ( x ) \cdot 1}{sin ( x ) sin ( x )}

Multiplicar:

cos (x) \cdot 1 = cos (x)

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

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

sin (x) sin (x)

Aplicar las leyes de los exponentes:

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

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

= sin^{1 + 1} (x)

Sumar:

1 + 1 = 2

= sin^{2} (x)

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

Factorizar

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

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

Factorizar

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

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

Factorizar el termino común

cos (x)

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

\frac{( - cos ^{2} ( x ) ) cos ( x )}{sin ^{2} ( x )} : - \frac{cos ^{3} ( x )}{sin ^{2} ( x )}

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

Quitar los parentesis:

(- a) = - a

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

- cos^{2} (x) cos (x) = - cos^{3} (x)

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

Aplicar las leyes de los exponentes:

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

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

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

Sumar:

2 + 1 = 3

= - cos^{3} (x)

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

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica básica:

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

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

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

Utilizar la identidad trigonométrica básica:

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

- cos (x) cot (x) cot (x)

Simplificar

- cos (x) cot (x) cot (x) : - cot^{2} (x) cos (x)

- cos (x) cot (x) cot (x)

Aplicar las leyes de los exponentes:

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

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

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

Sumar:

1 + 1 = 2

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

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Ejemplos populares

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

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

verificar (1+tan(x))/(sin(x))-sec(x)=csc(x)verificar

\frac{1 + tan ( x )}{sin ( x )} - sec (x) = csc (x)

verificar (csc(β))/(cot(β))=sec(β)verificar

\frac{csc ( β )}{cot ( β )} = sec (β)

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

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

verificar (csc(x)-cot(x))/(1-cos(x))=csc(x)verificar

\frac{csc ( x ) - cot ( x )}{1 - cos ( x )} = csc (x)