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verificar (1+sec(a))/(tan(a)+sin(a))=csc(a)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

\frac{1 + sec ( a )}{tan ( a ) + sin ( a )}

Expresar con seno, coseno

\frac{1 + sec ( a )}{sin ( a ) + tan ( a )}

Utilizar la identidad trigonométrica básica:

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

= \frac{1 + \frac{1}{c o s ( a )}}{sin ( a ) + tan ( a )}

Utilizar la identidad trigonométrica básica:

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

= \frac{1 + \frac{1}{c o s ( a )}}{sin ( a ) + \frac{s i n ( a )}{c o s ( a )}}

Simplificar

\frac{1 + \frac{1}{c o s ( a )}}{sin ( a ) + \frac{s i n ( a )}{c o s ( a )}} : \frac{1}{sin ( a )}

\frac{1 + \frac{1}{c o s ( a )}}{sin ( a ) + \frac{s i n ( a )}{c o s ( a )}}

Simplificar

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

en una fracción:

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

Simplificar

1 + \frac{1}{cos ( a )}

en una fracción:

\frac{cos ( a ) + 1}{cos ( a )}

1 + \frac{1}{cos ( a )}

Convertir a fracción:

1 = \frac{1 cos ( a )}{cos ( a )}

= \frac{1 \cdot cos ( a )}{cos ( a )} + \frac{1}{cos ( a )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{1 \cdot cos ( a ) + 1}{cos ( a )}

Multiplicar:

1 \cdot cos (a) = cos (a)

= \frac{cos ( a ) + 1}{cos ( a )}

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

Dividir fracciones:

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

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

Eliminar los terminos comunes:

cos (a)

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

Factorizar el termino común

sin (a)

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

Eliminar los terminos comunes:

cos (a) + 1

= \frac{1}{sin ( a )}

= \frac{1}{sin ( a )}

= \frac{1}{sin ( a )}

Re-escribir usando identidades trigonométricas

Utilizar la identidad trigonométrica básica:

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

\frac{1}{\frac{1}{c s c ( a )}}

Simplificar

\frac{1}{\frac{1}{c s c ( a )}}

Aplicar las propiedades de las fracciones:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{csc ( a )}{1}

Aplicar la regla

\frac{a}{1} = a

= csc (a)

csc (a)

csc (a)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

sin (4 x) = - 2 sin (2 x)

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

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

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

sin (2 t) = 2 sin (t) cos (t)