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Popular Trigonometría >

verificar (2sec^2(x)-2tan^2(x))/(csc(x))=sin(2x)sec(x)

Solución

verificar

\frac{2 sec ^{2} ( x ) - 2 tan ^{2} ( x )}{csc ( x )} = sin (2 x) sec (x)

Solución

Verdadero

Pasos de solución

\frac{2 sec ^{2} ( x ) - 2 tan ^{2} ( x )}{csc ( x )} = sin (2 x) sec (x)

Manipular el lado derecho

\frac{2 sec ^{2} ( x ) - 2 tan ^{2} ( x )}{csc ( x )}

Expresar con seno, coseno

\frac{2 sec ^{2} ( x ) - 2 tan ^{2} ( x )}{csc ( x )}

Utilizar la identidad trigonométrica básica:

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

= \frac{2 ( \frac{1}{c o s ( x )} ) ^{2} - 2 tan ^{2} ( x )}{csc ( x )}

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Aplicar las propiedades de las fracciones:

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

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

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

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

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

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

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

Multiplicar fracciones:

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

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

Multiplicar los numeros:

1 \cdot 2 = 2

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

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

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

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

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

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

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

Multiplicar fracciones:

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

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

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

Combinar las fracciones usando el mínimo común denominador:

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

Aplicar la regla

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

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

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

Quitar los parentesis:

(a) = a

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

Multiplicar

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

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

Multiplicar fracciones:

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

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

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

Aplicar las propiedades de las fracciones:

\frac{a}{1} = a

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Factorizar

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

2 - 2 sin^{2} (x)

Reescribir como

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

Factorizar el termino común

2

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

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

Eliminar los terminos comunes:

1 - sin^{2} (x)

= 2 sin (x)

= 2 sin (x)

= 2 sin (x)

Manipular el lado izquierdo

sin (2 x) sec (x)

Expresar con seno, coseno

sec (x) sin (2 x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

\frac{1}{cos ( x )} sin (2 x)

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot sin (2 x) = sin (2 x)

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Eliminar los terminos comunes:

cos (x)

= 2 sin (x)

= 2 sin (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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