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

Solución

verificar

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

Solución

Verdadero

Pasos de solución

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

Manipular el lado derecho

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )}

Expresar con seno, coseno

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{c o s ( θ )}{s i n ( θ )} - csc ( θ )}{\frac{c o s ( θ )}{s i n ( θ )} + csc ( θ )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{c o s ( θ )}{s i n ( θ )} - \frac{1}{s i n ( θ )}}{\frac{c o s ( θ )}{s i n ( θ )} + \frac{1}{s i n ( θ )}}

Simplificar

\frac{\frac{c o s ( θ )}{s i n ( θ )} - \frac{1}{s i n ( θ )}}{\frac{c o s ( θ )}{s i n ( θ )} + \frac{1}{s i n ( θ )}} : \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

\frac{\frac{c o s ( θ )}{s i n ( θ )} - \frac{1}{s i n ( θ )}}{\frac{c o s ( θ )}{s i n ( θ )} + \frac{1}{s i n ( θ )}}

Combinar las fracciones usando el mínimo común denominador:

\frac{cos ( θ ) + 1}{sin ( θ )}

Aplicar la regla

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

= \frac{cos ( θ ) + 1}{sin ( θ )}

= \frac{\frac{c o s ( θ )}{s i n ( θ )} - \frac{1}{s i n ( θ )}}{\frac{c o s ( θ ) + 1}{s i n ( θ )}}

Combinar las fracciones usando el mínimo común denominador:

\frac{cos ( θ ) - 1}{sin ( θ )}

Aplicar la regla

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

= \frac{cos ( θ ) - 1}{sin ( θ )}

= \frac{\frac{c o s ( θ ) - 1}{s i n ( θ )}}{\frac{c o s ( θ ) + 1}{s i n ( θ )}}

Dividir fracciones:

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

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

Eliminar los terminos comunes:

sin (θ)

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

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

Manipular el lado izquierdo

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Aplicar las propiedades de las fracciones:

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

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

Quitar los parentesis:

(a) = a

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

Cancelar

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

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

Factorizar

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

1 - 2 cos (θ) + cos^{2} (θ)

Escribir en la forma binómica

a x^{2} + b x + c

= cos^{2} (θ) - 2 cos (θ) + 1

Reescribir

cos^{2} (θ) - 2 cos (θ) + 1

como

cos^{2} (θ) - 2 cos (θ) \cdot 1 + 1^{2}

cos^{2} (θ) - 2 cos (θ) + 1

Reescribir

1

como

1^{2}

= cos^{2} (θ) - 2 cos (θ) + 1^{2}

Reescribir

2 cos (θ)

como

2 cos (θ) \cdot 1

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

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

Aplicar la formula del binomio al cuadrado:

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

a = cos (θ), b = 1

= (cos (θ) - 1)^{2}

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

Factorizar

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

1 - cos^{2} (θ)

Factorizar el termino común

- 1

= - (cos^{2} (θ) - 1)

Factorizar

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

cos^{2} (θ) - 1

Reescribir

1

como

1^{2}

= cos^{2} (θ) - 1^{2}

Aplicar la siguiente regla para binomios al cuadrado:

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

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

= (cos (θ) + 1) (cos (θ) - 1)

= - (cos (θ) + 1) (cos (θ) - 1)

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

Eliminar los terminos comunes:

cos (θ) - 1

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

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

Aplicar la regla

- (- a) = a

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Ejemplos populares

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

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

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

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

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

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

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