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verificar (1+csc(θ))/(sec(θ))=cos(θ)+cot(θ)

Solución

verificar

\frac{1 + csc ( θ )}{sec ( θ )} = cos (θ) + cot (θ)

Solución

Verdadero

Pasos de solución

\frac{1 + csc ( θ )}{sec ( θ )} = cos (θ) + cot (θ)

Manipular el lado izquierdo

cos (θ) + cot (θ)

Expresar con seno, coseno

cos (θ) + cot (θ)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

cos (θ) + \frac{cos ( θ )}{sin ( θ )} : \frac{cos ( θ ) sin ( θ ) + cos ( θ )}{sin ( θ )}

cos (θ) + \frac{cos ( θ )}{sin ( θ )}

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

Factorizar

\frac{cos ( θ ) + cos ( θ ) sin ( θ )}{sin ( θ )} : \frac{cos ( θ ) ( 1 + sin ( θ ) )}{sin ( θ )}

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

Factorizar

cos (θ) + cos (θ) sin (θ) : cos (θ) (1 + sin (θ))

cos (θ) + cos (θ) sin (θ)

Factorizar el termino común

cos (θ)

= cos (θ) (1 + sin (θ))

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

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

Re-escribir usando identidades trigonométricas

Utilizar la identidad trigonométrica básica:

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

\frac{( 1 + \frac{1}{c s c ( θ )} ) cos ( θ )}{\frac{1}{c s c ( θ )}}

Utilizar la identidad trigonométrica básica:

cos (x) = \frac{1}{sec ( x )}

\frac{( 1 + \frac{1}{c s c ( θ )} ) \frac{1}{s e c ( θ )}}{\frac{1}{c s c ( θ )}}

Simplificar

\frac{( 1 + \frac{1}{c s c ( θ )} ) \frac{1}{s e c ( θ )}}{\frac{1}{c s c ( θ )}}

Aplicar las propiedades de las fracciones:

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

= \frac{( 1 + \frac{1}{c s c ( θ )} ) \frac{1}{s e c ( θ )} csc ( θ )}{1}

Simplificar

1 + \frac{1}{csc ( θ )}

en una fracción:

\frac{csc ( θ ) + 1}{csc ( θ )}

1 + \frac{1}{csc ( θ )}

Convertir a fracción:

1 = \frac{1 csc ( θ )}{csc ( θ )}

= \frac{1 \cdot csc ( θ )}{csc ( θ )} + \frac{1}{csc ( θ )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{1 \cdot csc ( θ ) + 1}{csc ( θ )}

Multiplicar:

1 \cdot csc (θ) = csc (θ)

= \frac{csc ( θ ) + 1}{csc ( θ )}

= \frac{\frac{c s c ( θ ) + 1}{c s c ( θ )} \cdot \frac{1}{s e c ( θ )} csc ( θ )}{1}

Aplicar las propiedades de las fracciones:

\frac{a}{1} = a

= \frac{csc ( θ ) + 1}{csc ( θ )} \cdot \frac{1}{sec ( θ )} csc (θ)

Multiplicar fracciones:

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

= \frac{( csc ( θ ) + 1 ) \cdot 1 \cdot csc ( θ )}{csc ( θ ) sec ( θ )}

Eliminar los terminos comunes:

csc (θ)

= \frac{( csc ( θ ) + 1 ) \cdot 1}{sec ( θ )}

(csc (θ) + 1) \cdot 1 = csc (θ) + 1

(csc (θ) + 1) \cdot 1

Multiplicar:

(csc (θ) + 1) \cdot 1 = (csc (θ) + 1)

= (csc (θ) + 1)

Quitar los parentesis:

(a) = a

= csc (θ) + 1

= \frac{csc ( θ ) + 1}{sec ( θ )}

\frac{csc ( θ ) + 1}{sec ( θ )}

\frac{csc ( θ ) + 1}{sec ( θ )}

= \frac{1 + csc ( θ )}{sec ( θ )}

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Ejemplos populares

verificar (cos(x))/(sec(2x))-(sin(x))/(csc(2x))=cos(3x)verificar

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

verificar cot(a)+(sin(a))/(1+cos(a))=csc(a)verificar

cot (a) + \frac{sin ( a )}{1 + cos ( a )} = csc (a)

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sec^{2} (x) - cot^{2} (\frac{π}{2} - x) = 1

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

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sin (2 x) = - 2 sin (- x) cos (- x)