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verificar cot^2(x)-cos^2(x)cot^2(x)=cos^2(x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

cot^{2} (x) - cos^{2} (x) cot^{2} (x)

Expresar con seno, coseno

cot^{2} (x) - cos^{2} (x) cot^{2} (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

(\frac{cos ( x )}{sin ( x )})^{2} - cos^{2} (x) (\frac{cos ( x )}{sin ( x )})^{2} : \frac{cos ^{2} ( x ) - cos ^{4} ( x )}{sin ^{2} ( x )}

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

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

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

Aplicar las leyes de los exponentes:

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

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

cos^{2} (x) (\frac{cos ( x )}{sin ( x )})^{2} = \frac{cos ^{4} ( x )}{sin ^{2} ( x )}

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

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

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

Aplicar las leyes de los exponentes:

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

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

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

Multiplicar fracciones:

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

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

cos^{2} (x) cos^{2} (x) = cos^{4} (x)

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

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

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

= cos^{2 + 2} (x)

Sumar:

2 + 2 = 4

= cos^{4} (x)

= \frac{cos ^{4} ( x )}{sin ^{2} ( x )}

= \frac{cos ^{2} ( x )}{sin ^{2} ( x )} - \frac{cos ^{4} ( x )}{sin ^{2} ( x )}

Aplicar la regla

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

= \frac{cos ^{2} ( x ) - cos ^{4} ( x )}{sin ^{2} ( x )}

= \frac{cos ^{2} ( x ) - cos ^{4} ( x )}{sin ^{2} ( x )}

= \frac{cos ^{2} ( x ) - cos ^{4} ( x )}{sin ^{2} ( x )}

Re-escribir usando identidades trigonométricas

\frac{cos ^{2} ( x ) - cos ^{4} ( x )}{sin ^{2} ( x )}

Utilizar la identidad pitagórica:

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

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

= \frac{cos ^{2} ( x ) - cos ^{4} ( x )}{1 - cos ^{2} ( x )}

Simplificar

\frac{cos ^{2} ( x ) - cos ^{4} ( x )}{1 - cos ^{2} ( x )} : cos^{2} (x)

\frac{cos ^{2} ( x ) - cos ^{4} ( x )}{1 - cos ^{2} ( x )}

Factorizar

cos^{2} (x) - cos^{4} (x) : cos^{2} (x) (1 - cos^{2} (x))

cos^{2} (x) - cos^{4} (x)

Aplicar las leyes de los exponentes:

a^{b + c} = a^{b} a^{c}

cos^{4} (x) = cos^{2} (x) cos^{2} (x)

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

Factorizar el termino común

cos^{2} (x)

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

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

Eliminar los terminos comunes:

1 - cos^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

cot (\frac{1}{x}) = cos (\frac{1}{x}) arcsin (\frac{1}{x})

\frac{tan ( x )}{sin ( x )} - sec (x) = 0

\frac{sec ^{2} ( y )}{tan ( y )} = tan (y) + cot (y)

cos (2 0^{\circ}) = - sin^{2} (θ)

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