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

Soluzione

dimostrare

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

Vero

Fasi della soluzione

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

Manipolando il lato destro

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

Esprimere con sen e cos

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

Usare l'identità trigonometrica di base:

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}}

Semplifica

\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}}

Applica la regola degli esponenti:

(\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 )}}

Applica la regola degli esponenti:

(\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 )}}

Unisci

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

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

Converti l'elemento in frazione:

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 )}

Poiché i denominatori sono uguali, combinare le frazioni:

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

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

Moltiplicare:

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 )}}

Unisci

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

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

Converti l'elemento in frazione:

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 )}

Poiché i denominatori sono uguali, combinare le frazioni:

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

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

Moltiplicare:

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 )}}

Dividi le frazioni:

\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 ) )}

Cancella il fattore comune:

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 )}

Riscrivere utilizzando identità trigonometriche

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

Usa l'identità pitagorica:

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

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

Applicare la regola

\frac{a}{1} = a

= cos^{2} (β) - sin^{2} (β)

Usare l'Identità Doppio Angolo:

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

= cos (2 β)

= cos (2 β)

Abbiamo mostrato che i due lati possono prendere la stessa forma

\Rightarrow Vero

p ro v e tan (π + x) = tan (x)

p ro v e tan (A) = tan (A) csc^{2} (A) + cot (- A)

p ro v e \frac{1}{1 + sin ( x )} = (sec (x) - tan (x)) sec (x)

p ro v e tan (a) + cot (a) = \frac{2}{sin ( 2 a )}

p ro v e \frac{1 + sin ( α )}{cos ( α )} = \frac{cos ( α )}{1 - sin ( α )}