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verificar (2tan(x))/(1+tan^2(x))=sin(2x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

\frac{2 tan ( x )}{1 + tan ^{2} ( x )}

Expresar con seno, coseno

\frac{2 tan ( x )}{1 + tan ^{2} ( x )}

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Aplicar las leyes de los exponentes:

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

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

Multiplicar

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

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

Multiplicar fracciones:

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

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

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

Aplicar las propiedades de las fracciones:

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

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

Simplificar

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

en una fracción:

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

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

Convertir a fracción:

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

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

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

Multiplicar:

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

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

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

Multiplicar

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos (x)

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

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

Aplicar las propiedades de las fracciones:

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Utilizar la identidad pitagórica:

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

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

Aplicar la regla

\frac{a}{1} = a

= sin (2 x)

= sin (2 x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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