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verificar (cos^2(x)-1)(tan^2(x)+1)=-tan^2(x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

(cos^{2} (x) - 1) (tan^{2} (x) + 1)

Expandir

(- 1 + cos^{2} (x)) (1 + tan^{2} (x)) : - 1 - tan^{2} (x) + cos^{2} (x) + cos^{2} (x) tan^{2} (x)

(- 1 + cos^{2} (x)) (1 + tan^{2} (x))

Aplicar la propiedad distributiva:

(a + b) (c + d) = a c + a d + b c + b d

a = - 1, b = cos^{2} (x), c = 1, d = tan^{2} (x)

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

Aplicar las reglas de los signos

+ (- a) = - a

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

Simplificar

- 1 \cdot 1 - 1 \cdot tan^{2} (x) + 1 \cdot cos^{2} (x) + cos^{2} (x) tan^{2} (x) : - 1 - tan^{2} (x) + cos^{2} (x) + cos^{2} (x) tan^{2} (x)

- 1 \cdot 1 - 1 \cdot tan^{2} (x) + 1 \cdot cos^{2} (x) + cos^{2} (x) tan^{2} (x)

Multiplicar los numeros:

1 \cdot 1 = 1

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

Multiplicar:

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

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

Multiplicar:

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

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

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

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

Re-escribir usando identidades trigonométricas

- 1 + cos^{2} (x) - tan^{2} (x) + cos^{2} (x) tan^{2} (x)

Utilizar la identidad pitagórica:

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

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos^{2} (x)

= sin^{2} (x)

= - sin^{2} (x) - tan^{2} (x) + sin^{2} (x)

Sumar elementos similares:

- sin^{2} (x) + sin^{2} (x) = 0

= - tan^{2} (x)

= - tan^{2} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

7 csc^{2} (θ) - 5 cot^{2} (θ) = 2 csc^{2} (θ) + 5

cos (θ) = \frac{cot ( θ )}{csc ( θ )}

\frac{cos ( u )}{1 - sin ( u )} = sec (u) + tan (u)

cos (x - \frac{3 π}{2}) = - sin (x)

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