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verificar (cot^2(x))/(csc^2(x))=cos^2(x)

Solución

verificar

\frac{cot ^{2} ( x )}{csc ^{2} ( x )} = cos^{2} (x)

Verdadero

Pasos de solución

\frac{cot ^{2} ( x )}{csc ^{2} ( x )} = cos^{2} (x)

Manipular el lado derecho

\frac{cot ^{2} ( x )}{csc ^{2} ( x )}

Expresar con seno, coseno

\frac{cot ^{2} ( x )}{csc ^{2} ( x )}

Utilizar la identidad trigonométrica básica:

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

= \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{csc ^{2} ( x )}

Utilizar la identidad trigonométrica básica:

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

= \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{( \frac{1}{s i n ( x )} ) ^{2}}

Simplificar

\frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{( \frac{1}{s i n ( x )} ) ^{2}} : cos^{2} (x)

\frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{( \frac{1}{s i n ( x )} ) ^{2}}

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

= \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{\frac{1}{s i n ^{2} ( x )}}

(\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{\frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}{\frac{1}{s i n ^{2} ( x )}}

Dividir fracciones:

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

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

Simplificar

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

Eliminar los terminos comunes:

sin^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

(cos (x) - sin (x)) 2 = 1 - 2 sin (x) cos (x)

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

sin (x) cos (x) csc^{2} (x) = cot (x)

4 cos^{2} (x) - 2 = 0

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