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verificar-sin^2(x)sec^2(x)=1-sec^2(x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

- sin^{2} (x) sec^{2} (x)

Re-escribir usando identidades trigonométricas

- sin^{2} (x) sec^{2} (x)

Utilizar la identidad pitagórica:

sec^{2} (x) = tan^{2} (x) + 1

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

Utilizar la identidad pitagórica:

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

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

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

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

Expandir

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

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

Expandir

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

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

Aplicar la propiedad distributiva:

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

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

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

Aplicar las reglas de los signos

+ (- a) = - a

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

Simplificar

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

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

Multiplicar los numeros:

1 \cdot 1 = 1

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

Multiplicar:

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

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

Multiplicar:

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

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

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

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

Poner los parentesis

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

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

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

tan^{2} (x) + 1 = sec^{2} (x)

- tan^{2} (x) - 1 = - sec^{2} (x)

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

Utilizar la identidad trigonométrica básica:

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

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

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)

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

Utilizar la identidad pitagórica:

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

= 1 - sec^{2} (x)

= 1 - sec^{2} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

sin^{2} (θ) + sin^{2} (θ) tan^{2} (θ) = tan^{2} (θ)

1 + sin (2 θ) = (cos (θ) sin (θ))^{2}

sec (θ) csc (θ) = tan (θ) + cot (θ)

2 tan (0^{\circ}) = sin (0^{\circ})