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verificar tan^2(A)(csc^2(A)-1)=1

Solución

verificar

tan^{2} (A) (csc^{2} (A) - 1) = 1

Verdadero

Pasos de solución

tan^{2} (A) (csc^{2} (A) - 1) = 1

Manipular el lado derecho

tan^{2} (A) (csc^{2} (A) - 1)

Expresar con seno, coseno

(- 1 + csc^{2} (A)) tan^{2} (A)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

(\frac{1}{sin ( A )})^{2} = \frac{1}{sin ^{2} ( A )}

(\frac{1}{sin ( A )})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{1 ^{2}}{sin ^{2} ( A )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( A )}

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

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

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

Aplicar las leyes de los exponentes:

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

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

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

Multiplicar fracciones:

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

= \frac{sin ^{2} ( A ) ( - 1 + \frac{1}{s i n ^{2} ( A )} )}{cos ^{2} ( A )}

Simplificar

- 1 + \frac{1}{sin ^{2} ( A )}

en una fracción:

\frac{- sin ^{2} ( A ) + 1}{sin ^{2} ( A )}

- 1 + \frac{1}{sin ^{2} ( A )}

Convertir a fracción:

1 = \frac{1 sin ^{2} ( A )}{sin ^{2} ( A )}

= - \frac{1 \cdot sin ^{2} ( A )}{sin ^{2} ( A )} + \frac{1}{sin ^{2} ( A )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- 1 \cdot sin ^{2} ( A ) + 1}{sin ^{2} ( A )}

Multiplicar:

1 \cdot sin^{2} (A) = sin^{2} (A)

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

= \frac{\frac{- s i n ^{2} ( A ) + 1}{s i n ^{2} ( A )} sin ^{2} ( A )}{cos ^{2} ( A )}

Multiplicar

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

sin^{2} (A) \frac{- sin ^{2} ( A ) + 1}{sin ^{2} ( A )}

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

sin^{2} (A)

= - - sin^{2} (A) + 1

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Aplicar la regla

\frac{a}{a} = 1

= 1

= 1

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

cos^{3} (θ) = - 1

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

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

(tan (x)) (sin (x)) + (cos (x)) = sec (x)

cos (k (x) x) = cos (- k (x) x)