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Popular Trigonometría >

verificar (tan(x)+cot(x))/(sec(x)csc(x))=1

Solución

verificar

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

Solución

Verdadero

Pasos de solución

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

Manipular el lado derecho

\frac{tan ( x ) + cot ( x )}{sec ( x ) csc ( x )}

Expresar con seno, coseno

\frac{cot ( x ) + tan ( x )}{csc ( x ) sec ( x )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{c o s ( x )}{s i n ( x )} + tan ( x )}{csc ( x ) sec ( x )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}{csc ( x ) sec ( x )}

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

\frac{\frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}{\frac{1}{s i n ( x )} \cdot \frac{1}{c o s ( x )}} : cos^{2} (x) + sin^{2} (x)

\frac{\frac{c o s ( x )}{s i n ( x )} + \frac{s i n ( x )}{c o s ( x )}}{\frac{1}{s i n ( x )} \cdot \frac{1}{c o s ( x )}}

Multiplicar

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

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

Multiplicar fracciones:

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

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

Multiplicar los numeros:

1 \cdot 1 = 1

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

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

Simplificar

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

en una fracción:

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

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

Mínimo común múltiplo de

sin (x), cos (x) : sin (x) cos (x)

sin (x), cos (x)

Mínimo común múltiplo (MCM)

Calcular una expresión que este compuesta de factores que aparezcan tanto en

sin (x)

cos (x)

= sin (x) cos (x)

Reescribir las fracciones basandose en el mínimo común denominador

Multiplicar cada numerador por la misma cantidad necesaria para multiplicar el denominador correspondiente y convertirlo en el mínimo común denominador

Para

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

multiplicar el denominador y el numerador por

cos (x)

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

Para

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

multiplicar el denominador y el numerador por

sin (x)

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

= \frac{\frac{c o s ^{2} ( x ) + s i n ^{2} ( x )}{s i n ( x ) c o s ( x )}}{\frac{1}{s i n ( x ) c o s ( x )}}

Dividir fracciones:

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

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

Simplificar

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

Eliminar los terminos comunes:

sin (x)

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

Eliminar los terminos comunes:

cos (x)

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

= 1

= 1

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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