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verificar cos(2β)=(1-tan^2(β))/(1+tan^2(β))

Solución

verificar

cos (2 β) = \frac{1 - tan ^{2} ( β )}{1 + tan ^{2} ( β )}

Verdadero

Pasos de solución

cos (2 β) = \frac{1 - tan ^{2} ( β )}{1 + tan ^{2} ( β )}

Manipular el lado izquierdo

\frac{1 - tan ^{2} ( β )}{1 + tan ^{2} ( β )}

Expresar con seno, coseno

\frac{1 - tan ^{2} ( β )}{1 + tan ^{2} ( β )}

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar las leyes de los exponentes:

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

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

Simplificar

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

en una fracción:

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

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

Convertir a fracción:

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

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

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

Multiplicar:

1 \cdot cos^{2} (β) = cos^{2} (β)

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

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

Simplificar

1 - \frac{sin ^{2} ( β )}{cos ^{2} ( β )}

en una fracción:

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

1 - \frac{sin ^{2} ( β )}{cos ^{2} ( β )}

Convertir a fracción:

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

= \frac{1 \cdot cos ^{2} ( β )}{cos ^{2} ( β )} - \frac{sin ^{2} ( β )}{cos ^{2} ( β )}

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} ( β ) - sin ^{2} ( β )}{cos ^{2} ( β )}

Multiplicar:

1 \cdot cos^{2} (β) = cos^{2} (β)

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

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

Dividir fracciones:

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

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

Eliminar los terminos comunes:

cos^{2} (β)

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

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

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

Re-escribir usando identidades trigonométricas

\frac{cos ^{2} ( β ) - sin ^{2} ( β )}{cos ^{2} ( β ) + sin ^{2} ( β )}

Utilizar la identidad pitagórica:

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

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

Aplicar la regla

\frac{a}{1} = a

= cos^{2} (β) - sin^{2} (β)

Utilizar la identidad trigonométrica del ángulo doble:

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

= cos (2 β)

= cos (2 β)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

sec (x) - tan (x) = \frac{1}{sec ( x )}

tan^{2} (x) + cot^{2} (x) = sec^{2} (x) csc^{2} (x)

tan^{4} (x) = \frac{( sin ^{4} ( x ) )}{cos ^{4} ( x )}

sin^{2} (π) = 1 - cos^{2} (π)